Geometry of Incompatible Deformations: Differential Geometry in Continuum Mechanics 9783110563214, 9783110562019

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Sergey Lychev and Konstantin Koifman Geometry of Incompatible Deformations

De Gruyter Studies in Mathematical Physics

| Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, São Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia

Volume 50

Sergey Lychev and Konstantin Koifman

Geometry of Incompatible Deformations | Differential Geometry in Continuum Mechanics

Physics and Astronomy Classification Scheme 2010 Primary: 46.05.+b; Secondary: 02.40.Yy Authors Prof. Sergey Lychev Ishlinsky Institute for Problems in Mechanics RAS (IPMech RAS) [email protected] Konstantin Koifman Department of Applied Mathematics Bauman Moscow State Technical University [email protected]

ISBN 978-3-11-056201-9 e-ISBN (PDF) 978-3-11-056321-4 e-ISBN (EPUB) 978-3-11-056227-9 ISSN 2194-3532 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: CPI books GmbH, Leck www.degruyter.com

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Preface This book is intended to provide a systematic treatment of those parts of modern differential geometry that are essential for the modeling of incompatible finite deformations in solids. Included are discussions of generalized deformation and stress measures on smooth manifolds, geometrical formalization for structurally inhomogeneous bodies, various definitions for material connection, and evolution equations for them. The incompatibility of deformations may be caused by a variety of physical phenomena; among them are distributed dislocations and disclinations, point defects, non-uniform thermal fields, shrinkage, growth, etc. Incompatible deformations result in residual stresses and distortion of geometrical shape. These factors are associated with critical parameters in modern high-precision technologies, particularly, in additive manufacturing, and are considered to be ipso facto essential constituents in the corresponding mathematical models. In this context, the development of methods for their quantitative description is the actual problem of modern solid mechanics. The methods in question are based on the representation of a body and physical space in terms of differentiable manifolds, namely material manifold and physical manifold. These manifolds are equipped with specific metrics and connections, non-Euclidian in general. The book is divided into 14 chapters. The first one is an introduction. The second one briefly reviews various representations of the geometry of physical space and time, including Euclidean, Minkowski, and curved space time models. This review, on the one hand, leads to exhaustive definition of physical space in terms of smooth manifolds. On the other hand, it draws attention to the specific aspects of geometrical formalization for space and space time, which have much in common with their counterparts in the modeling of elastic bodies, such as smooth (material) manifolds. The third chapter focuses on the essentials of conventional non-linear elasticity. In this chapter, those fundamentals of continuum mechanics that are rooted in the concept of absolute, Euclidean, space and absolute, Newtonian, time are discussed. Special attention is paid to the assumption that global stress-free shapes of any body exists. Rejection of this assumption reflects the main idea of the present book. Common notations for strain and stress measures, variational symmetries, and constitutive equations, which would be generalized in the rest of the book, are highlighted. In the fourth chapter, we consider the issue of the physical interpretation of the non-Euclidean structure of the material manifold. It is shown that a two-dimensional rigid surface, which formalizes the curved substrate used in the deposition process, may serve as an example of a non-Euclidean physical manifold. Affine connection on the material manifold represents the intrinsic properties (inner geometry) of the body and is determined by the field of local uniform configurations performing its “assembly” of identical and uniform infinitesimal “bricks”. Uniformity means that the response functional gives the same response on all admissible smooth deformahttps://doi.org/10.1515/9783110563214-201

VIII | Preface

tions for them. As a result of assembling, one obtains a body that cannot be immersed in an undistorted state into the physical manifold. It is an essential feature of residual stressed bodies produced by additive processes. For this reason, it should benefit from immersion into a non-Euclidean space (material manifold with non-Euclidean material connection). To this end, it is convenient to formalize the body and physical space in terms of the theory of smooth manifolds. The deformation is formalized as embedding (or, in a special case, as immersion) of the former manifold into the latter one. The fifth chapter is dedicated to generalization of relations for Cauchy–Green strain measures. They are generated by embedding of a Riemannian manifold, representing the body, into a Riemannian manifold, representing the space. We consider such issues as the transpose of the deformation gradient and generalization of the Cauchy decomposition theorem on smooth manifolds. The sixth chapter covers the definition of motion, velocity, and acceleration fields in the framework of the theory of smooth manifolds. The motion is represented as a time-dependent flow. The seventh chapter addresses the issues of the formalization of stress and power measures by fields defined on smooth manifolds that represent the body and physical space. It contains the systematic construction of a theory for the general nonEuclidean case. Forces are interpreted as covectors, i.e., as a linear functionals, whose action on the velocity vectors of material points results in mechanical power. The abstract theory of integration based on exterior form formalism is adapted to the elements of this structure, which allows one to formulate the power balance equation of the material manifold (similarly to the reference description in the classical mechanics of compatible deformation) and of the physical one (similarly to the spatial description). In the eighth chapter the response of hyperelastic solids on smooth manifolds is generalized. Only simple materials are considered: their response depends on the local configuration and material points. For such materials, the notion of material isomorphism is introduced. The main assumption that leads to the notion of a structurally inhomogeneous solid is the following: for each body of simple material, there exists a family of configurations, which index set is identical to the set of material points constituting the whole body. Each configuration maps its index point to the uniform state. This assumption is used for synthesizing the material metric. A method is proposed for describing a deformable body of variable composition as a family of Riemannian manifolds over which partition and joining operations are defined. These operations characterize the structural features of the inhomogeneities given by the additive process scenario. In the ninth chapter, the various ways to specify the general form of affine connections on the material and physical manifolds are considered. Affine connection endows manifolds with geometric properties, in particular, with parallel translation rules for vector fields. For simple materials the parallel translation is an elegant math-

Preface | IX

ematical formalization of the concept of a materially uniform (in particular, stressfree) non-Euclidean reference shape. In fact, one can obtain a connection on physical space by determining the parallel transport rule as a transformation of the tangent vector, which corresponds to the structure of the physical space containing shapes of the body. In turn, one can obtain affine connection on material manifold by defining a parallel transport rule as the transformation of the tangent vector, in which its inverse image with respect to locally uniform embeddings does not change. Utilizing the conception of material connections and the corresponding methods of non-Euclidean geometry may significantly simplify formulation of the initial boundary value problems of the theory of incompatible deformations, so the choice of connections is important. Connection on the physical manifold is compatible with the metric, and the Levi–Civita relation holds for it. Connection of the material manifold is considered in two alternative variants. The first leads to the Weitzenböck space (the space of absolute parallelism or teleparallelism, i.e., space with zero curvature and non-metricity but with non-zero torsion) and gives a clear interpretation of the material connection in terms of the local linear transformations that return an elementary volume of simple material to the uniform state. The second one leads to the Riemannian space (space with zero non-metricity and torsion but with non-zero curvature). Examples for such generalizations are considered in the rest of the chapter. The tenth chapter refers to the balance equations, which are obtained in terms of Cartan’s exterior covariant derivative from the general principle of covariance. The eleventh chapter contains the examples of inhomogeneous solids, which inhomogeneity was induced by some additive technological process. Various types of evolutionary problems, owning to different technological regimes, are considered. The calculations are illustrated by numerical computations and graphs. In the last chapters, the required mathematical preliminaries adapted to the present book are considered. May 2018, Moscow

Sergey Lychev Konstantin Koifman

Contents Preface | VII General Scheme of Notations | XIX 1

Introduction | 1

2 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.1.8 2.1.9 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.2.9 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5

Geometry of Physical Space | 3 Euclidean Affine Space | 4 General Definitions | 4 Topology on E | 9 Smooth Mappings on E | 9 Parallel Transport | 13 Total Derivatives of Vector and Tensor Fields | 14 Integration | 16 Symmetrization and Antisymmetrization | 17 Divergence | 17 Curl | 18 Curvilinear Coordinates | 19 Reparametrization | 19 Coordinate Curves | 21 Local Basis | 21 From Pointwise to the Whole | 22 Metric Tensors | 23 Parallel Transport | 24 Total Derivatives | 24 Integration | 28 The Physical Basis | 28 Nabla Operator Formalism in 3D-Space | 28 Definitions of Nabla | 29 Operations in Euclidean Space | 29 Nabla and Mappings | 30 Coordinate Representations for Divergence | 32 Coordinate Representations for Curl | 33 Riemannian Space | 33 General Definition | 33 Smooth Mappings on P | 35 Geometric Measures and Geodesics | 37 Parallel Transport | 38 Integration | 40

XII | Contents

2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.6 2.6.1 2.6.2 2.6.3 2.6.4 2.6.5 2.6.6 2.6.7 2.7

Newtonian Space-Time | 40 Newtonian Space-Time Manifold | 41 Newton’s Laws | 42 Rigid Frame | 44 Change of Frame | 44 Relativistic Space-Time | 46 Lorentzian Manifolds | 46 Time Orientation | 47 Definition of Relativistic Space-Time | 48 Observers | 48 Lorentz Transformations | 49 Matter | 49 The Einstein Equation | 51 Concluding Remarks | 57

3 Essentials of Non-Linear Elasticity Theory | 59 3.1 Shapes and Deformation | 59 3.2 Shape Coordinates and Basis | 61 3.3 Deformation Gradient and Strain Measures | 64 3.3.1 Deformation Gradient | 64 3.3.2 Strain Measures | 66 3.3.3 Deformation Gradient in Curvilinear Coordinates | 67 3.3.4 Strain Measures in Curvilinear Coordinates | 68 3.4 Displacement Field | 69 3.5 Motion | 70 3.6 Compatibility Conditions | 70 3.6.1 Review on de Rham Cohomology | 70 3.6.2 Necessary and Sufficient Conditions for Compatibility | 71 3.7 Stresses | 75 3.8 Non-Linear Elasticity as Field Theory | 78 3.8.1 Action and Its Lagrangian | 78 3.8.2 Partial and Full Variations | 79 3.8.3 Field Equations | 84 3.8.4 Action Invariance Conditions | 84 3.9 Constitutive Relations | 87 3.9.1 Principle of Material-Frame Indifference | 87 3.9.2 The Cauchy Polar Decomposition Theorem | 89 3.9.3 Simple Material | 90 3.9.4 Representation Theorems | 92 3.10 Hyperelastic Solids | 94 3.10.1 Expressions for Stresses | 94 3.10.2 Universal Deformations | 94

Contents |

3.11 3.11.1 3.11.2 3.11.3 3.11.4 3.12 3.12.1 3.12.2 3.12.3 3.12.4 3.12.5 3.12.6 3.12.7 3.12.8 3.12.9 3.12.10 4

XIII

Linearized Elasticity | 103 Linearized Kinematics | 103 Linearized Constitutive Relations | 105 Linearized Stresses | 106 The Green–Rivlin–Shield–Truesdell Formula | 107 Distributed Defects in Solids | 109 Preliminary Remarks | 109 Total Distortion 1-Forms | 110 Four-Dimensional Space and the Homotopy Operator | 111 Decomposition of Total Distortion into Exact and Antiexact Parts | 112 Continuity Equations of Defect Dynamics | 112 4D Representation | 113 The Momentum Equation | 113 Equations in Matrix Form | 114 Relations of Dislocation and Disclination Forms with Connection, Curvature, and Torsion Forms | 114 Application of Yang–Mills Coupling Theory | 116

4.3 4.4 4.5 4.5.1 4.5.2 4.5.3 4.5.4

Geometric Formalization of the Body and Its Representation in Physical Space | 119 Geometric Motivation | 119 Comparison between Conventional and Non-Euclidean Continuum Mechanics | 125 A Body | 128 Configurations | 132 A Shape of a Body as a Submanifold of the Physical Space | 135 A Shape as a Submanifold | 135 The Local k-slice Condition | 136 The Induced Riemannian Space Structure | 137 A Shape and the Physical Space. Intrinsic versus Spatial | 137

5 5.1 5.2 5.3 5.3.1 5.3.2 5.4 5.4.1 5.4.2 5.5 5.6

Strain Measures | 141 Review on Cauchy Theory | 141 Configurations and Deformations | 143 Coordinate Representations of Configurations and Deformations | 144 The General Case | 144 The Euclidean Case | 146 Two-Point Tensors | 149 Two-Point Tensor Bundle | 149 The Transpose and Orthogonal Tensors | 151 Configuration Gradient | 153 Left and Right Cauchy–Green Strain Tensors | 156

4.1 4.2

XIV | Contents

5.6.1 5.6.2 5.6.3

Spatial Measurements in Material Description | 156 The Cauchy Polar Decomposition Theorem | 157 Cauchy–Green Strain Measures as Pullback and Pushforward of Metrics | 158

6 6.1 6.2 6.3 6.4 6.4.1 6.4.2 6.4.3 6.4.4 6.5

Motion | 161 Motion as a Curve | 161 Velocity | 162 (m + 1)-Formalism and Acceleration | 164 Flows and Lie Derivatives | 165 Vector Fields and Integral Curves | 165 Flow | 166 Lie Derivatives | 168 Time-Dependent Flow | 174 Motion as a Time-Dependent Flow | 176

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Stress Measures | 179 Concentrated Forces and Force Densities | 179 Inclined Hyperplanes | 181 Piola Section on Inclined Hyperplanes | 183 The Cauchy Section | 185 Cauchy and Piola Stresses | 186 Transformation from Spatial Description to Material | 188 Example: Neo-Hookean Solids | 191 The Case dim B = dim P = 3 | 192 The Eshelby Energy-Momentum Tensor on Manifolds | 193

8 8.1 8.2 8.3 8.4 8.5 8.5.1 8.5.2 8.5.3

Material Uniformity and Inhomogeneity | 195 Equivalence Relation Between Smooth Embeddings | 195 Local Configurations and Simple Bodies | 195 Material Uniformity | 196 A Material Metric | 199 Bodies with Variable Material Composition | 200 Formalization of Bodies with Variable Material Composition | 200 The Discrete Process | 203 The Continuous Process | 204

9 9.1 9.2 9.2.1 9.2.2

Material Connections | 209 Connections on Vector Bundles | 209 Affine Connection | 212 The Transformation Law | 213 Torsion, Curvature, and Non-Metricity | 214

Contents | XV

9.2.3 9.2.4 9.2.5 9.2.6 9.2.7 9.2.8

A Particular Case: Euclidean Space | 216 A Particular Case: Riemannian Space | 218 Connection on the Pullback Bundle | 220 The Moving Frame Method | 222 The Weitzenböck Connection as a Material Connection | 225 The Weitzenböck Connection: Example | 226

10 Balance Equations | 231 10.1 Divergence | 231 10.1.1 The Case of a Vector Field | 231 10.1.2 The Case of a Covector-Valued Form | 234 10.2 The Reynolds Transport Theorem | 239 10.3 Balance Equations in Integral Form | 240 10.3.1 Power Balance | 240 10.3.2 Mass Conservation | 242 10.3.3 Transformation of the Spatial Power Balance Equation | 243 10.4 Derivation of Differential Balance Equations Using the Covariance Principle | 244 10.4.1 The Principle of Covariance | 244 10.4.2 Change of Frame and Objective Transformations | 245 10.4.3 Derivation of Spatial Conservation Laws | 246 11 The Evolutionary Problem – Examples | 249 11.1 Example: The Cylindrical Problem | 249 11.1.1 Hollow Cylinders with Discrete Inhomogeneity | 249 11.1.2 Hollow Cylinders with Continuous Inhomogeneity | 255 11.1.3 Results and Discussion | 263 11.2 Uniform Inflation of a Spherical Multilayered Structure | 267 11.2.1 Layers and Assemblies | 267 11.2.2 Coordinates and Vector Bases | 268 11.2.3 Strain Measures | 271 11.2.4 Incompressible Material | 271 11.2.5 Compressible Material | 273 11.2.6 Continuous Non-Euclidean Structures | 277 11.2.7 Small Perturbations of the Self-Stressed Shape | 280 11.3 Bending of Rectangular Blocks | 281 11.3.1 Deformations of a Single Block | 281 11.3.2 Stresses | 286 11.3.3 Forces on Boundary Surfaces | 287 11.3.4 Thin Layers | 289 11.3.5 Discrete Accretion | 290 11.3.6 Continuous Accretion | 292

XVI | Contents

12 Algebraic Structures | 297 12.1 Preliminary Comments on the Use of Sets | 297 12.2 Ordered Pairs. Cartesian Products. Relations | 297 12.3 Functions | 298 12.4 Some Algebraic Structures | 300 12.4.1 Groups | 300 12.4.2 Ring | 300 12.4.3 Module | 301 12.5 Linear Spaces and Mappings | 301 12.5.1 Vector Space over ℝ | 301 12.5.2 Linear and k-Linear Mappings | 303 12.5.3 Tensor Products of Vector Spaces | 305 12.5.4 Vectors and Linear Mappings in Euclidean Space | 308 12.6 Linear Groups | 311 12.7 Affine Space | 312 13 Review of Smooth Manifolds and Vector Bundles | 315 13.1 Smooth Manifolds | 315 13.1.1 Topological Spaces | 315 13.1.2 Smooth Structure | 321 13.1.3 Smooth Mappings | 322 13.1.4 Embedded Submanifolds | 325 13.2 The “Tower” of Tensor Spaces | 326 13.2.1 Tangent Space to a Smooth Manifold | 326 13.2.2 The tangent Map at a Point | 332 13.2.3 Cotangent Space to a Smooth Manifold | 333 13.2.4 Remarks on the Tangent Spaces | 334 13.2.5 The “Tower” | 336 13.2.6 Exterior Forms | 337 13.3 Vector Bundles and Their Sections | 340 13.3.1 Smooth Vector Bundles of Rank k | 340 13.3.2 Tangent and Cotangent Bundles | 341 13.3.3 Operations on Vector Bundles | 343 13.3.4 Vector Bundles of Higher Rank | 344 13.3.5 Sections of Vector Bundles | 345 13.3.6 Vector Bundle Homomorphisms | 347 13.3.7 Pullback and Pushforward | 348 13.3.8 Smooth Frames | 350 13.3.9 Exterior Differentiation | 351 13.3.10 The Riemannian Metric and Musical Isomorphisms | 352 13.4 Orientation and Integration on Manifolds | 354 13.4.1 The Volume Form and Orientation of Smooth Manifolds | 354

Contents | XVII

13.4.2 13.4.3

The Hodge Star Operator | 355 Integration of Differential Forms and Stokes’ Theorem | 356

14 Connections on Principal Bundles | 359 14.1 Lie Groups and Lie Algebras | 359 14.1.1 Lie Groups and Homomorphisms | 359 14.1.2 Group Action | 359 14.1.3 Lie Algebra of the Lie Group | 361 14.1.4 Adjoint Representation | 363 14.1.5 Exponential Mapping | 363 14.2 Principal Bundles | 364 14.2.1 Bundles | 364 14.2.2 Principal Bundles | 365 14.2.3 Frame Bundles | 366 14.2.4 Associated Bundles | 367 14.3 Connections | 368 14.3.1 Connections on the Principal Bundle | 368 14.3.2 Local Representation of Connections | 369 14.3.3 Local Representation on Frame Bundles | 369 14.3.4 Gauge Maps | 370 14.3.5 Parallel Transport | 371 14.3.6 Curvature | 374 14.3.7 Torsion | 375 14.3.8 Bianchi Identities | 376 14.3.9 Covariant Derivatives on Associated Vector Bundles | 376 14.3.10 Direct Construction of Covariant Derivatives on Principal Bundles | 377 Bibliography | 381 Index | 387

General Scheme of Notations Index of frequently used symbols Symbol

Name

Place of definition or first occurrence

a, a [a ij ] A, A A B, B, B i j

Spatial acceleration Matrix with elements a ij Material acceleration Action Left Cauchy–Green tensor Body Right Cauchy–Green tensor Set of all configurations The algebra of all smooth functions on M The space of all smooth covector fields on M Cartesian coordinate map Euclidean affine space Deformation gradient Configuration gradient Spatial metric Material metric General linear group of degree s Inclusion map of X Orthonormal basis Identity map of X The set of all invertible linear mappings from V to W Identity operator Jacobian of deformation Local configuration at point X Vector space of linear mappings from V to W Orthogonal group in dimension s Position vector field First Piola–Kirchhoff covector-valued form Set of all parts of a body B First Piola–Kirchhoff stress tensor Abstract physical space Part of a body Nonmetricity Rank of vector bundle E Riemann curvature tensor The space of all smooth sections of vector bundle E The set of all symmetric linear mappings from V to V Cauchy covector-valued form Cauchy stress tensor

Section 6.3 Section 2.2.3 Section 6.3 Section 2.6.6 Section 3.3 Section 4.3 Section 3.3 Section 4.4 Section 13.1.3 Section 13.3.5 (2.2) Section 2.1.1 Section 3.3 Section 5.5 Section 2.7 Section 4.3 Section 12.6 (13.1) Section 2.1.1 (12.1) Section 12.5.2 Section 3.4 (3.8) Section 8.2 Section 12.5.2 Section 12.6 (2.1) (7.3) Section 4.3 Section 3.7 Section 2.7 Section 4.3 (9.10) Section 13.3.1 (9.9) Section 13.3.5 Section 12.5.4 (7.4) Section 3.7

B C, C, C ij C(B; P) C ∞ (M) CVec(M) D E F F g G GL(s; ℝ) ιX (i s )m s=1 IdX Inv(V; W) I J KX , K X Lin(V; W) O(s) p P Part(B) P P P Q rank(E) R Sec(E) Sym(V) T T

https://doi.org/10.1515/9783110563214-202

XX | General Scheme of Notations

Symbol

Name

Place of definition or first occurrence

T Tp M T p∗ M u v, v V, V V Vec(M) γ i jk , Γ ijk Λ k (T p∗ M) Ω k (M)

Torsion Tangent space to M at p ∈ M Cotangent space to M at p ∈ M Displacement vector field Spatial velocity Material velocity Translation vector space The space of all smooth vector fields on M Affine connection coefficients The set of all exterior k-forms The set of all differential k-forms on M

(9.8) Section 13.2.1 Section 13.2.3 Section 3.4 Section 6.2 Section 6.2 Section 2.1.1 Section 13.3.5 Section 9.2 Section 13.2.6 Section 13.3.5

Index of frequently used operations Operation

Name

Place of definition or first occurrence

⋅ ⌟, ⌞ [⋅, ⋅] ⊗ ∧ ∧̇ d D, d

Inner product Contractions Lie bracket Tensor product Wedge (Exterior) product Inner-exterior product Exterior derivative Cartan’s exterior derivative

Section 12.5.4 Section 13.3.5 (13.20) Section 12.5.3 Section 13.2.6 Section 10.1.2 Section 13.3.9 Section 10.1.2

1 Introduction Geometrical modeling methods of continuum mechanics are rooted in such parts of modern differential geometry as exterior calculus, theory of connections, theory of G-structures, etc. [1–3]. The overarching purpose of this book is to illustrate applications of geometrical methods in the modeling of structurally inhomogeneous solids. This kind of inhomogeneity appears in uniform solids due to incompatibility of local finite deformations. One can find the application of the proposed methods to the design of modern additive technologies, which is a rich field of research [4]. The elegance and effectiveness of geometrical language in continuum physics have been demonstrated in some excellent and comprehensive works [5–8]. The use of this language has a long history. The pioneering works of E. Beltrami, H. Poincaré, T. Levi-Civita, etc. [9] at the end of the nineteenth century laid the foundation for the application of non-Euclidean geometry in physics and mechanics. At the beginning of the twentieth century, A. Einstein, H. Weyl, H. Cartan, and others [10] used the geometric language in the framework of the general theory of relativity [11]. This gave impetus to the development of relativistic non-linear continuum mechanics. Relativistic models of deformable continua were obtained in terms of Riemannian and non-Riemannian geometry [12–15]. Herewith, the physical space containing a deformable body was equipped with a non-Euclidian structure. The idea that the space, which contains the reference, unstressed shape of the body, should have a non-Euclidean structure was first proposed by Eckart¹ [16] and Kondo [17–20], although the first studies of incompatible deformations in solids go back to Weingarten [21] and Volterra [22]. Notice that the physical space containing the observed (actual) shape has a Euclidean structure. Speaking figuratively, a body from an imaginary non-Euclidean space is mapped into a physical, Euclidean space. This causes deformations of a special kind, which can be referred to as intrinsic (or local) deformations, since they do not disappear in the absence of external force fields. Kondo [20] demonstrated that such deformations fully characterize the fields of defects as sources of residual stresses. These sources, in turn, can be determined by specific, material, connection, and corresponding parallel transport in the space containing the reference shape. The material connection can be fully characterized by the relevant curvature and torsion. In a series of classical works, J. Nye, K. Kondo, E. Kröner, and R. De Wit [18–20, 23, 24] demonstrated the effectiveness of the geometrical approach to the modeling of crystals with distributed defects (dislocations, disclinations, point defects, etc.) The

1 Carl Eckart, 1948: «The traditional theory of the solid state rests on two false assumptions. One is the principle of a constant relaxed (or standard) state. The other is the principle of relaxability-in-thelarge, first formulated mathematically by de Saint-Venant. His equations are essentially identical with Riemann’s equations expressing the condition that a geometry be Euclidean-in-the-large». https://doi.org/10.1515/9783110563214-001

2 | 1 Introduction

main feature of this approach lies in the specific geometry of space that represents the body itself. In general, this geometry is non-Euclidean. The torsion, curvature, and non-metricity of the connection, which determine its non-Euclidean properties, are closely related to the densities of distributed defects. Geometrical methods have been applied successfully to solve non-linear problems in the theories of surface growth and volumetric growth [25]. A few words should be said about the relevance of models for bodies with intrinsic deformations. At present, there is a variety of manufacturing processes that allow us to create a solid, especially thin films, by using layer-by-layer (LbL) technologies [26]. Stereolithography, electrochemical deposition, welding, etc. [4, 27] are among these. Such methods allow us to produce a body with a complicated shape that can be manufactured in non-conventional ways only. The essential mechanical feature of solids, produced by additive technologies, is that they always have intrinsic deformations. For example, such a phenomenon arises as a consequence of polymer shrinkage due to solidification. It leads to residual stresses and distortion of shape. In addition, it should be noted that non-linear effects manifest themselves significantly in additive technological processes. This is related to the high concentration of energy in the sintering zone (in laser welding processes) or to the complicated rheology of photochemical solidification processes. Therefore, general models should be formulated within non-linear approaches of continuum mechanics. The typical feature of bodies generated during some additive process is the absence of a stress-free shape in the conventional sense. From the physical point of view, this means that the body stores elastic energy that cannot be released under any smooth deformation. This energy can be released only by cutting the body into an infinite number of disjoined parts. A similar situation arises in the theory of distributed defects (dislocations, disclinations, pores) [28]. When a stress-free shape does not exist, the specific mechanical and physical features of the body can be quantitatively represented in a variety of ways, for example, by measuring the deformation incompatibility [24], or the density of the distributed defects, or the intensity of residual stresses [29]. Such notions are closely related and can be described within the framework of the general idea of deformation as a general geometric transformation in nonEuclidian space. Lastly, it should be noted that the topological properties of the body can be essentially non-trivial; in particular, the body may have the structure of a non-orientable surface, for instance, a Klein surface. The “physical” implementation of such a body can be represented by a self-intersecting grid shell [30]. In light of the rapid development of additive manufacturing technologies, such “exotic” bodies can be easily implemented in the material.

2 Geometry of Physical Space A theme running through this book is the proposition that the incompatible local deformations, arising in elastic solids for a variety of reasons, can be mathematically modeled as an embedding of a smooth manifold, representing a body, into a smooth manifold, representing a physical space. In order to take an advantage of such formalism, the general guidelines for the geometrical description of actual objects should be defined. Our point of departure will be the review of various alternatives for space and time formalization. The mathematical structure of space-time is the subject of geometrodynamics, a science that studies the fundamentals of space, time, and gravitation. The reader may find a comprehensive view of these issues in [31–34]. This book is not meant to be consistent presentation of geometrodynamics and only uses some of its results to show the affinity of geometric approaches in theoretical physics and non-linear elasticity. The present chapter is aimed to set the stage for the definition of physical space in terms of a smooth manifold. In this regard, we discuss in detail some particular ways for the mathematical description of flat and curved two- and three-dimensional physical space as well as four-dimensional space-time.A flat three-dimensional model for physical space determines a relatively simple (but not at all elementary) formal structure that contains elements that can be considered as prototypes of more sophisticated ones, defined in non-Euclidean spaces. Apart from this, it is intended to serve as an arena for the embodiment of three-dimensional bodies in the form of their shapes within the framework of classical non-linear elasticity. Curved two-dimensional (Riemannian) space provides an opportunity for the modeling of material surfaces and curvilinear membranes. Curved four-dimensional spacetime gives the example of a more general formal structure that has much in common with the non-Euclidian structure of the material manifold discussed in Chapter 4. In this book, we systematically use diagrams that illustrate relations between mappings, their domains, and codomains¹. For example, if A, B, C, D are sets and f : A → B, g : B → C, h : A → D, j : D → C are mappings, then the diagram A

f

B g

h D

j

C

illustrates the relations between the sets and the maps. Such a diagram is called commutative if g ∘ f = j ∘ h.

1 A more detailed outline of the theory of mappings is given in Chapter 12, Section 12.1. https://doi.org/10.1515/9783110563214-002

4 | 2 Geometry of Physical Space

We use ordered n-tuples, which consist of sets and mappings, for defining geometric structures such as affine space, Riemannian space, etc. If it does not lead to any misunderstanding, after the first reference to such ordered tuples we omit its components and simply write the underlying set.

2.1 Euclidean Affine Space The naive definition of Euclidean space comes down to the definition of the space of “directed line segments”. Technically, it can be given in terms of a collection of mutually perpendicular unit length directed segments, namely unit vectors. Herein, the notion of inner product has already been inherent in pure geometric properties of unit vectors. However, such an approach usually neglects the distinction between points and translation vectors (radius-vectors) that bring the origin to those points. In the following, we apply a more rigorous approach based on Weyl axioms. In the framework of this approach, the points and translation vectors are treated as elements of different sets, whose properties and relations are axiomatically defined. This would enable us to determine the structure of classical flat physical space in more detail and to undertake a more comprehensive comparison of flat space with curved space. We would like to highlight that space, according to [35], is “a form of existence of the real world”. The experience gained in the terrestrial (local) laboratory suggests the axiomatics for this form, which was outlined in general terms by Greek geometers [36]. It is synthetic geometry, the key tool of Greek mathematics, that leads to the modern formulation of affine-Euclidean space. It is based on the principle of congruence, which establishes the equivalence of geometric objects, i.e., figures, and bodies, which can be transformed one into the other by a combination of rigid motions, namely a translation, a rotation, and a reflection. Together, these motions form a Euclidean group of isometries. We begin with a general definition.

2.1.1 General Definitions Affine space. Affine space is a triplet (E, V, ψ), where² (i) E is a set whose elements are referred to as spatial points; (ii) V is a real vector space whose elements are referred to as translation vectors; (iii) ψ : E × E → V is a mapping, which assigns to each ordered pair (a, b) ∈ E × E 󳨀→ some vector from V (denoted by ab or b − a), such that

2 Here we briefly introduce the notions that are connected with affine space. The definition of affine space and a listing of its properties are presented in Chapter 12, Section 12.7.

2.1 Euclidean Affine Space |

5

W1 ) for every points a, b, c ∈ E the following relation holds: 󳨀→ 󳨀 → 󳨀 → ab + bc + ca = 0 ∈ V ; W2 ) for any point a ∈ E and for any vector v ∈ V there exists a unique point 󳨀→ b ∈ E, such that ab = v. The set of two axioms W1 ) and W2 ) is usually associated with the name of H. Weyl [35]. The relation in axiom W1 ) is associated with the name Chasles or with the “parallelogram rule”. According to axiom W2 ), for any fixed point a ∈ E the mapping 󳨀→ ψ a (x) := ax ,

ψa : E → V ,

is a bijection. This allows one to define the external operation +: E × V → E ,

+ : (a, v) 󳨃→ a + v := ψ−1 a (v) ,

󳨀→ which assigns to each ordered pair (a, v) a unique point b ∈ E, such that ab = v. Dimension of affine space. The dimension of (E, V, ψ) is defined as the dimension of V, i.e., dimE := dim V = m. In the following, only finite dimensions will be considered. Affine coordinate system. An affine coordinate system can be determined by an origin o ∈ E and a basis (e i )m i=1 for V. It is represented by a doublet (o, (e 1 , . . . , e m )). According to W2 ), the mapping p: E → V ,

p(x) := x − o ,

(2.1)

is a bijection. In mechanics, the vector p(x) is known primarily as the position vector (radius-vector or place vector) of a point x [37–39]. Euclidean affine space. Euclidean affine space is an 4-tuple (E, V, (⋅), ψ), where (E, V, ψ) is an affine space and (⋅) is an inner product on V (see Chapter 12, Section 12.5.4). Rectilinear coordinates. In order to obtain numerical values for the components of the position vector one has to define the dual basis (e i )m i=1 for V from the system of i i linear equations e ⋅e j = δ j , i, j = 1, . . . , m. Then, any vector u ∈ V can be represented as the decomposition u = u k e k , where u k = e k ⋅ u. Thus, ∀x ∈ E :

x = o + xk ek ,

where

x k = e k ⋅ p(x), k = 1, . . . , m .

We refer to the m-tuple (x i )m i=1 as rectilinear coordinates. Rectilinear coordinates can be represented as images of the bijective mapping D(o,(ei )mi=1 ) : E → ℝm ,

D(o,(ei )mi=1 ) (x) := (e1 ⋅ p(x), . . . , e m ⋅ p(x)) ,

(2.2)

6 | 2 Geometry of Physical Space which is completely determined by the (m + 1)-tuple (o, (e 1 , . . . , e m )). The m-tuple m (x k )m k=1 = D(o,(ei )i=1 ) (x) represents coordinates of a point x relative to (o, (e 1 , . . . , e m )). We refer to the map D(o,(ei )mi=1 ) as a Cartesian coordinate map. In the following, we omit the lower index of D(o,(ei )mi=1 ) , since we do not consider different affine coordinates simultaneously. The relations between D and p are illustrated in the following commutative diagram: p E V −1

D ℝm

Φ

=

p

∘D

In the diagram, Φ : V → ℝm is a vector space isomorphism. Cartesian coordinates. A particular case of the affine coordinate system is the Cartesian coordinate system (o, (i 1 , . . . , i m )), where (i k )m k=1 is an orthonormal basis of V. This basis is defined by the relations i k ⋅ i j = δ kj , k, j = 1, . . . , m. For any vector u ∈ V, one has u = u k i k , where u k = i k ⋅ u. Multivectors. The inner product allows one to calculate lengths and angles. At the same time, for volume measurements one needs some extra tool. Such a tool is provided by the wedge (exterior) product ∧ on vectors and is intimately related with the notion of the determinant. This product has the following properties³: (i) ∀u, v ∈ V : u ∧ v = −v ∧ u; (ii) ∀u, v, w ∈ V ∀a, b ∈ ℝ: u ∧ (av + bw) = au ∧ v + bu ∧ w ; (iii) ∀u, v, w ∈ V : (u ∧ v) ∧ w = u ∧ (v ∧ w). For example, let m = 2 and (o, (i 1 , i2 )) be a Cartesian system. In classical synthetic geometry, the area is defined as the specific property of the rectangular Π0 with vertices o, o + i1 , o + i2 , o + i1 + i2 . Consider the “elementary” bivector i1 ∧ i2 as a basic (reference) object with the geometric sense of the unit area, i.e., the unit area of Π0 . Now let u, v ∈ V be vectors. Consider the parallelogram Π with vertices o, o + u, o + v, o + u + v. If one has u = u k i k , v = v k i k , then the value of the determinant 󵄨󵄨 1 󵄨u ∆(u, v) = 󵄨󵄨󵄨󵄨 2 󵄨󵄨u

󵄨 v1 󵄨󵄨󵄨 󵄨󵄨 2 v 󵄨󵄨󵄨

3 Here we present the sketch of multivector and wedge product notions. A comprehensive treatment can be found in [40, 41].

2.1 Euclidean Affine Space |

7

gives the dimensionless oriented area of Π. The wedge product (bivector) u ∧ v is defined as u ∧ v := ∆(u, v)i1 ∧ i2 , and gives the oriented area of Π. Developing the example, consider the case m = 3. Let (o, (i 1 , i 2 , i3 )) be a Cartesian system. Then, one has three “elementary” bivectors i2 ∧ i3 , i1 ∧ i3 , and i1 ∧ i2 that give mutually perpendicularly oriented unit areas of the corresponding parallelograms. Again, let u, v ∈ V be vectors and Π the parallelogram, like in the preceding example. Then, for u = u k i k , v = v k i k , the square root 󵄨󵄨 2 √󵄨󵄨󵄨󵄨u 󵄨󵄨u 3 󵄨

󵄨2 󵄨󵄨 1 v2 󵄨󵄨󵄨 󵄨u 󵄨󵄨 + 󵄨󵄨󵄨 3 󵄨󵄨u v3 󵄨󵄨󵄨 󵄨

󵄨2 󵄨󵄨 1 v1 󵄨󵄨󵄨 󵄨u 󵄨󵄨 + 󵄨󵄨󵄨 2 󵄨󵄨u v3 󵄨󵄨󵄨 󵄨

󵄨2 v1 󵄨󵄨󵄨 󵄨󵄨 v2 󵄨󵄨󵄨

is the dimensionless area of Π. Define bivector u ∧ v as 󵄨 󵄨 󵄨󵄨 1 󵄨󵄨 1 󵄨󵄨 2 v2 󵄨󵄨󵄨 v1 󵄨󵄨󵄨 󵄨󵄨u 󵄨󵄨u 󵄨u 󵄨 󵄨 󵄨 󵄨󵄨 2 ∧ i + ∧ i + u ∧ v := 󵄨󵄨󵄨󵄨 3 i i 2 3 1 3 󵄨 󵄨 󵄨 3 3 3 󵄨 󵄨 󵄨 󵄨󵄨u v 󵄨󵄨 v 󵄨󵄨 󵄨󵄨u 󵄨󵄨u 󵄨

󵄨 v1 󵄨󵄨󵄨 󵄨󵄨 i1 ∧ i2 . v2 󵄨󵄨󵄨

The geometric sense of u ∧ v is, again, the oriented area of Π. To the bivector u ∧ v there corresponds the vector u × v, namely, the cross-product of u and v. This vector is defined as having the same components as the bivector, i.e., u × v := e ijk u i v j i k ,

(2.3)

where e ijk is the alternator:

e ijk = e

ijk

1, { { { := {−1, { { { 0,

if (i, j, k) is an even permutation of (1, 2, 3) , if (i, j, k) is an odd permutation of (1, 2, 3) ,

(2.4)

if (i, j, k) is not a permutation of (1, 2, 3) .

Similarly to the notion of area, in classical synthetic geometry, the volume is defined as the specific property of the parallelepiped P0 with vertices o, o + i1 , o + i2 , o + i1 + i2 , o + i3 , o + i1 + i3 , o + i2 + i3 and o + i1 + i2 + i3 . The “elementary” three-vector i1 ∧ i2 ∧ i3 is defined as a basic (reference) object with the geometric sense of the unit volume, i.e., the unit volume of P0 . Now, let u, v, w ∈ V. Consider the parallelepiped P with vertices o, o + u, o + v, o + u + v, o + w, o + u + w, o + v + w and o + u + v + w. Its dimensionless oriented volume is given by 󵄨󵄨 1 󵄨 v1 w1 󵄨󵄨󵄨 󵄨󵄨u 󵄨󵄨 2 󵄨󵄨 V(u, v, w) = 󵄨󵄨󵄨 u v2 w2 󵄨󵄨󵄨 , 󵄨󵄨 3 󵄨󵄨 v3 w3 󵄨󵄨󵄨 󵄨󵄨󵄨 u where u = u k i k , v = v k i k , w = w k i k . Define the three-vector u ∧ v ∧ w as u ∧ v ∧ w := V(u, v, w)i 1 ∧ i2 ∧ i3 . Thus, u ∧ v ∧ w is the oriented volume of P.

8 | 2 Geometry of Physical Space

Generalization of the notions of bivector and three-vector lead to the notion for multivector [40]. The examples considered illustrate the close relations between multivectors, determinants, and volume measurements. That is, the inner product (⋅) and wedge product ∧ represent the complete set of tools for metric measurements on E. Remark 2.1. Above we defined the Euclidean affine space axiomatically and we will systematically use the axiomatic approach to define more general spaces further. However, it is important to note that, since for the last two thousand years the structure of Euclidean space has been studied in detail within the framework of synthetic (Greek) geometry, it becomes possible to construct a structure, equivalent to Euclidean space equipped with a Cartesian coordinate system, relying only on the concept of congruence. Referring again to the argument of H. Weyl [35], we show a way of a synthetic (coordinate-free) reasoning that leads to the idea of a Cartesian space. The primary concepts of synthetic geometry are a point, a curve, and a surface. The congruence principle allows one to choose the straight lines of all the curves by the following condition: straight lines coincide with their images under arbitrary translations along themselves and, thus, can be defined by any two distinct points on them. Such translations correspond to rotations and screw motions of the ambient space, which gives a kinematic interpretation of the straight line as the rotation axes. Any point on a line divides it into three parts, the point and two open half-lines. These half-lines are congruent, because they can be superimposed on each other by movement (rotation around the point). Two different half-lines define an angle; among all the angles is a special angle that is congruent with its complementary one. This is a right angle, and the continuation of its sides are perpendicular straight lines (note that the notion of perpendicularity is introduced here without using the metric, in spirit of the arguments of synthetic geometry). The collection of all lines passing through a given point a and perpendicular to a given line l, containing this point, defines a new geometrical object, namely the plane Π la , which has rotational symmetry with respect to l. The translation of the plane Π la along the line l preserves the property of its perpendicularity to the line l. Therefore, one can obtain the family {Π lα }α∈l of planes parallel to Π l . Choosing a straight line j lying in some Π lα , and repeating the procedure described above, we obtain a family β of planes Π j that are perpendicular to the planes Π lα . Finally, choosing a line k lying β γ α󸀠 in Π l , α 󸀠 ∈ l and in some Π j , β ∈ j, we obtain the third family Π k which elements are β perpendicular to the elements of family Π lα and family Π j . The pairwise intersections of such planes are straight lines. These lines can be “graded” by means of a sequence of translations of unit intervals i, j, and k, which can be aligned with each other by rotations. As a result, we obtain spatial forests of graded lines, which constructively define the Cartesian net and the corresponding coordinate system. Bearing in mind the fact that later we will move on from affine Euclidean space to spaces with a more general geometry, let us explain how the advanced analytical fea-

2.1 Euclidean Affine Space | 9

tures that usually are pointed out in the definition of general smooth manifolds can be defined in the Euclidean approach.

2.1.2 Topology on E The Weyl axioms, establishing the relation between E and V, as well as the algebraic structure of the Euclidean space V, define a particular kind of topology on E, namely, the metric topology. Let us explain this in more detail. E as a metric space. The inner product (⋅) induces the norm ‖ ⋅ ‖ on V: ∀u ∈ V : ‖u‖ := √u ⋅ u . The norm (2.5), in turn, induces a metric on E as follows: 󳨀→ d : E × E → ℝ , d(a, b) := ‖ab‖ .

(2.5)

(2.6)

Thus, E is endowed with the metric topology⁴. The mapping D (2.2) is a homeomorphism relatively to this topology and the topology of ℝm . The metric topology turns E into Hausdorff topology space (E, S) with basis B ⊂ S that is defined by the collection of open balls with rational radii and centers: ̃ ∃α ∈ ℚ ∀a ∈ A : ‖a − o‖ < α) , (A ∈ B) ⇔ (∃o ∈ E ̃ = D−1 (ℚm ) is a dense countable subset of E. Since B is countable, the topowhere E logical space E is second countable. Atlas on (E, S). The ordered pair (E, D) represents a chart on E. Such a chart covers the whole E, and the minimal atlas Aaffine = {(E, D)} is trivial. Thus, E is a smooth manifold with smooth structure generated by Aaffine .

2.1.3 Smooth Mappings on E The affine-Euclidean space, which we presented as a topological space (E, S) equipped with the trivial atlas Aaffine , can be used to define a smooth manifold, if we define a smooth structure on it. To this end, we introduce the notions of smooth⁵ curves, functions, and fields as follows. Curves. Define a curve on E as a mapping χ : 𝕀 → E, where 𝕀 ⊂ ℝ is an interval. A curve χ has the coordinate representation ̃χ = D ∘ χ : 𝕀 → ℝm relatively to the atlas Aaffine . The curve χ is said to be smooth (belongs to class C∞ ) if the mapping ̃χ is smooth, i.e., ̃χ ∈ C∞ (𝕀; ℝ). 4 Definitions of a metric, a topology, and related notions are given in Chapter 13, Section 13.1.1. 5 Smooth mappings between manifolds are briefly considered in Chapter 13, Section 13.1.3.

10 | 2 Geometry of Physical Space Functions. Define a function on E as a mapping f : O → ℝ, where O ⊂ E is an open set. Its coordinate representation has the form: ̃f = f ∘ D−1 : D(O) → ℝ. That is, ̃f is a real valued function of m real arguments. The function f is said to be smooth (belongs to class C∞ ) if the mapping ̃f is smooth, i.e., ̃f ∈ C∞ (D(O); ℝ). Vector fields. A vector field on E is a mapping u : O → V, where O ⊂ E is an open set. It can be represented relatively to Aaffine as a vector valued function of m real arguments: ̃ = u ∘ D−1 : D(O) → V , u or as a mapping

̃̃ = Φ ∘ u ∘ D−1 : D(O) → ℝm , u

where Φ = D ∘ p−1 : V → ℝm is a vector space isomorphism. The singleton Avect = {(V, Φ)} is a trivial atlas on V and, thus, V is a smooth manifold with smooth structure generated by Avect . The smoothness of a vector field u is defined by the smooth̃̃ relatively to the atlases Aaffine and Avect . ness of its coordinate representation u The following commutative diagrams illustrate relations between functions, vector fields, and their coordinate representations: O D D(O)

f

ℝ

̃f

O D D

u

̃u ̃̃ u

V Φ ℝm

Covectors. The affine-Euclidean space possesses all the facilities for calculating the components of translation vectors with respect to the chosen basis. Indeed, they can be found as a result of a generally non-orthogonal projection that is analytically determined by a scalar product with elements of a dual basis. Moreover, the scalar product defines the concept of the angle and, together with the definition of vector product, represents the complete framework of elementary geometry. Thus, the requirements of the geometer are completely satisfied. However, this cannot be said about the requirements of a physicist! In order to determine pair vector-like objects, for example, energetically conjugate velocity-force pairs, one has to consider the space of linear functionals (covectors) over V, namely, dual space V ∗ . The dual space satisfies all the axioms of the vector space (see Chapter 12, Section 12.5.1), and, moreover, if the dimension of V is finite, then the dimension of V ∗ coincides with it (for proof of this assertion, see [42]). It should be noted that in certain cases, covectors without loss of generality can be represented by vectors dual to them. However, such an identification of pair objects can lead to a loss of physical meaning. For this reason, we will further strictly distinguish between vectors and covectors.

2.1 Euclidean Affine Space |

11

More detailed definitions are given as follows. The elements of V ∗ := Lin(V; ℝ), the dual vector space of V, are linear functionals over V (covectors). The dimension of dual space is equal to the dimension of V. The value of a covector ν ∈ V ∗ on a vector u ∈ V is often written as ⟨ν, u⟩ := ν(u). It is convenient to represent covectors ∗ in the basis (e i )m i=1 for V that satisfy the property ⟨e i , e j ⟩ = δ ij ,

i, j = 1, . . . , m .

The m-tuple (e i )m i=1 is a dual basis in the dual space. In the following we will refer to it in a shorter manner as a dual basis. In the dual basis, covectors can be represented by their components: ∀ν ∈ V ∗ ∃ν1 , . . . , ν m ∈ ℝ :

ν = νk ek .

The norm ‖ ⋅ ‖ on V induces the norm on V ∗ as follows: ∀ν ∈ V ∗ :

‖ν‖ := sup |ν(u)| . ‖u‖=1

V∗

With such a norm the dual space is a smooth manifold; its smooth structure is generated by the trivial atlas Acovect = {(V ∗ , Ψ)}. Here, Ψ : V ∗ → ℝm ,

Ψ : ν = ν k e k 󳨃→ (ν1 , . . . , ν m ) ,

is the vector space isomorphism. Remark 2.2. Despite the fact that in classic works on continuum mechanics, the distinction between vectors and covectors is rarely taken into account, distinguishing them as objects from different spaces seems to be useful from the physical standpoint. Indeed, if all quantities that possess a vector nature are defined in the same vector space, which, as a rule, is equipped with a scalar product, then a completely justified desire arises to treat, for example, the velocity vector and the force vector⁶ as elements of one vector space. Their scalar product results in a scalar (a power). On the other hand, since they belong to the same space, one can formally consider their sum, which, of course, has no physical meaning. This incorrect situation can be redressed by the following consideration: the force had to be formalized as a covector, while the velocity naturally has to be represented by a vector. The covector (a force), as a linear functional, acts on the velocity vector, which results in a scalar (a power). Of course, the sum of a vector and any covector is not defined. It should be noted that the difference in vector-like quantities, such as force and displacement, was already discussed in the pioneering works on vector and tensor calculus [35]. 6 The concept of force is one of the fundamentals in modern physics. However, this concept does not have a strict and universal definition. This fact was described figuratively by Max Jammer in his monograph [43]: To introduce the term “force” as an explanatory element in the theory of physical science means to develop a misleading vocabulary.

12 | 2 Geometry of Physical Space Musical isomorphisms. Recall that V is equipped with scalar product (⋅). According to the Riesz theorem⁷ for any covector ν ∈ V ∗ there exists the unique vector w ∈ V, such that ∀u ∈ V : ν(u) = w ⋅ u . The mapping (⋅)♯ : V ∗ → V, which assigns to each covector ν ∈ V ∗ the corresponding (due to the Riesz theorem) vector (ν)♯ = w ∈ V, is a vector space isomorphism. Its inverse is denoted by (⋅)♭ : V → V ∗ . Thus, for a vector v ∈ V, one has ∀u ∈ V : v ♭ (u) = v ⋅ u . The mutually inverse isomorphisms (⋅)♭ and (⋅)♯ are referred to as musical isomorphisms⁸. Covector fields. A covector field on E is a mapping ν : O → V ∗ , where O ⊂ E is an open set. Relatively to the atlas Aaffine , a covector field ν is represented by ̃ν = ν ∘ D−1 : D(O) → V ∗ . In atlases Aaffine and Acovect one has the following representation: ̃̃ν = Ψ ∘ ν ∘ D−1 : D(O) → ℝm . The smoothness of a covector field ν is defined by the smoothness of its coordinate representation ̃̃ν. Tensor fields. Tensors are elements of tensor products of vector spaces⁹ V, V ∗ . Thus, mixed tensors of type (k, l) are elements of V ⊗ ⋅ ⋅ ⋅ ⊗ V ⊗ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ V∗ ⊗ ⋅ ⋅ ⋅ ⊗ V∗ . T (k,l)(V) := ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ k copies

The field ℝ, the

vector spaces V, and V ∗ T (0,0) (V) = ℝ ,

l copies

are particular cases of mixed tensor spaces:

T (1,0) (V) = V ,

T (0,1) (V) = V ∗ .

Equivalently, a (k, l)-type tensor can by viewed as a multilinear map, the element of ∗ , . . . , V ∗ , V, . . . , V ; ℝ) . Link+l (V ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ k copies

l copies

Thus, one can introduce a norm on T (k,l)(V) as follows: ‖T‖ :=

sup

‖ν 1 ‖=1,...,‖ν k ‖=1, ‖u1 ‖=1,...,‖ul ‖=1

|T(ν1 , . . . , ν k , u 1 , . . . , u l )| ,

7 The formulation and proof can be found in Chapter 12, Section 12.5.4. 8 These isomorphisms are introduced in Chapter 12, Section 12.5.4. Their generalization for smooth manifolds is given in Chapter 13, Section 13.3.10. 9 The formal definition of an abstract tensor product is given in Chapter 12, Section 12.5.3.

2.1 Euclidean Affine Space | 13

i m for each T ∈ T (k,l) (V). A basis (e i )m i=1 and its dual, (e )i=1 , establish vector space isok+l (k,l) m (V) → ℝ , morphism Θ : T

Θ: T = T

i 1 ...i k

j1 ...j l e i 1

⊗ ⋅ ⋅ ⋅ ⊗ e i k ⊗ e j1 ⊗ ⋅ ⋅ ⋅ ⊗ e j l 󳨃→ (T

i 1 ...i k

j1 ...j l )1≤i ,...,i ≤m ,1≤j ,...,j ≤m 1 k 1 l

.

The vector space T (k,l) (V) is endowed with the smooth manifold structure. The smooth structure is generated by the trivial atlas Atensor = {(T (k,l) (V), Θ)}. A (k, l)-type tensor field on E is a map T : O → T (k,l) (V), where O ⊂ E is an open set. Relatively to the atlas Aaffine , a tensor field T is represented by ̃ T = T ∘ D−1 : D(O) → T (k,l) (V) . In atlases Aaffine and Atensor , one has the following representation: k+l ̃̃ T = Θ ∘ T ∘ D−1 : D(O) → ℝm .

As in the previous cases, the smoothness of a tensor field T is defined by smoothness ̃̃ of its coordinate representation T.

2.1.4 Parallel Transport The concept of parallel transport plays a crucial role in geometrodynamics, as well as in the geometrical theory of incompatible deformations. In this regard, special attention will be given to this concept within the present book. The most simple definition for it is given in the Euclidean geometry. The inner product (⋅) on V allows one to introduce the cosine mapping cos : V \ {0} × V \ {0} → [−1, 1] ,

cos(u 1 , u 2 ) :=

u1 ⋅ u2 . ‖u 1 ‖‖u 2 ‖

(2.7)

The value of cos(u 1 , u 2 ) on a pair (u 1 , u 2 ) of vectors has the sense of a cosine of angle between them. The mapping cos allows one to introduce the notion of parallelism as follows. Vectors u, v ∈ V are said to be parallel, if | cos(u 1 , u 2 )| = 1. A vector field u : O → V is said to be weakly parallel transported along a curve χ : [a, b] → O if ∀x ∈ [a, b] :

| cos(u(χ(a)), u(χ(x)))| = 1 ,

and is said to be strongly parallel transported if ∀x ∈ [a, b] :

u(χ(a)) = u(χ(x)) .

The condition of weakly parallel transport is equal to the representation u(x) = f(x)u(χ(a)), for all x ∈ χ([a, b]). Here, f : χ([a, b]) → ℝ is a function.

14 | 2 Geometry of Physical Space

2.1.5 Total Derivatives of Vector and Tensor Fields In order to apply calculus one has to introduce the notion of derivatives for vector and tensor fields. We begin with the following definition. Total derivative of a function. Suppose that O ⊂ E is an open set and f : O → ℝ is a function. It is called differentiable at a point a ∈ O if there exists a covector ν a ∈ V ∗ , such that for any h ∈ V, a + h ∈ O, the following identity holds [44]: f(a + h) = f(a) + ⟨ν a , h⟩ + φ(h)‖h‖ ,

(2.8)

where limh→0,a+h∈O φ(h) = 0. In terms of the E. Landau symbol o, one can write f(a + h) = f(a) + ⟨ν a , h⟩ + o(‖h‖)

as

h→0.

The mapping ν a is called the total derivative of f at point a, and since it is the unique mapping that satisfies (2.8), one of the following notations can be used: Df(a), f 󸀠 (a). The uniqueness follows from the fact that the value of grad f(a) is the directional derivative: f(a + th) − f(a) ∀h ∈ V : ⟨f 󸀠 (a), h⟩ = lim , O a;h ∋t→0 t where O a;h = {t ∈ ℝ | t ≠ 0, a + th ∈ O}. If f is differentiable at each point x ∈ O of the set O, then the mapping f 󸀠 : O → V∗ ,

x 󳨃→ f 󸀠 (x)

is defined correctly and is called the total derivative of f . Gradient. Due to Riesz theorem there exists the unique vector grad f(a) := (f 󸀠 (a))♯ ,

(2.9)

such that ⟨f 󸀠 (a), h⟩ = h ⋅ grad f(a). The vector grad f(a) is called the gradient f at point a. i m ∗ 󸀠 In the basis (e i )m i=1 of V and the dual basis (e )i=1 of V , the covector f (a) has the representation: f 󸀠 (a) = ∂ i f(a)e i ,

where

∂ i f(a) = ⟨f 󸀠 (a), e i ⟩ ,

i = 1, . . . , m .

Total derivative of a vector field. Suppose that O ⊂ E is an open set and u : O → V is a vector field. It is called differentiable at a point a ∈ O, if there exists a linear map (a (1, 1)-type tensor) L a ∈ Lin(V; V), such that for any h ∈ V, a + h ∈ O, the following identity holds [44]: u(a + h) = u(a) + L a [h] + o(‖h‖)

as

h→0.

(2.10)

The mapping L a is called the total derivative of u at the point a, and since it is the unique mapping that satisfies (2.10), the following notations are used: Du(a), u 󸀠 (a).

2.1 Euclidean Affine Space | 15

As in the case of a function, the uniqueness follows from the fact that the value of Du(a) is the directional derivative: ∀h ∈ V :

Du(a)[h] =

lim

O a;h ∋t→0

u(a + th) − u(a) , t

where O a;h = {t ∈ ℝ | t ≠ 0, a + th ∈ O}. If u is differentiable at each point x ∈ O, then the mapping Du :

O → Lin(V; V) ,

x 󳨃→ Du(x)

is defined correctly and is called the total derivative of u. In the basis (e i )m i=1 of V, the vector field u has the representation: ∀x ∈ O :

u(x) = u i (x)e i .

Here the mappings u i : O → ℝ, i = 1, . . . , m, are component functions. The differentiability of u at a ∈ O implies the differentiability of each u i at a. In the basis (e i )m i=1 ∗ , the linear map Du(a) has the representation: and the dual basis (e i )m of V i=1 Du(a) = ∂ j u i (a)e i ⊗ e j ,

where

∂ j u i (a) = ⟨Du i (a), e j ⟩ ,

i, j = 1, . . . , m .

The total derivative of a covector field, and, generally, of a tensor field is defined in an analogous way. In all such cases, the smoothness in terms of the total derivative coincides with the smoothness in terms of smooth manifolds. The smoothness of a vector, covector, and, generally, tensor fields is equivalent to the smoothness of all its component functions. Remark 2.3. We considered the cases of a function and vector field separately. That allowed us to pick out individual features of their derivatives. In analysis, the notion of the total derivative (Fréchet derivative) at a point is introduced in a uniform way as follows [44]. Suppose that (A1 , V1 , ψ1 ) and (A2 , V2 , ψ2 ) are finite dimensional affine spaces with translation spaces V1 and V2 , respectively¹⁰. That is, A1 , A2 are sets, V1 , V2 are finite dimensional real vector spaces and ψ1 : A1 ×A1 → V1 , ψ2 : A2 ×A2 → V2 are mappings that satisfy the Weyl axioms W1 ) and W2 ). In addition, suppose that translation spaces V1 and V2 are endowed with norms ‖ ⋅ ‖1 and ‖ ⋅ ‖2 . These norms turn A1 and A2 into metric spaces by defining metrics on them in the same manner as in Section 2.1.2 of the present chapter. Let X ⊂ A1 be a non-empty set and a ∈ X be an interior point¹¹ of X. A mapping f : X → A2 is called differentiable at the point a, if there exists a linear map La ∈ Lin(V1 ; V2 ), such that for any h ∈ V1 , a + h ∈ X, the following identity holds¹²: f(a + h) = f(a) + La [h] + φ(h)‖h‖1 ,

(2.11)

10 Recall that the definition of affine space and its properties are given in Chapter 12, Section 12.7. 11 That is, a is a center of some open ball in X. 12 Here, one needs to use the external operations +i : Ai × Vi → Ai , i = 1, 2, which are constructed by virtue of the axiom W2 ). For readability we omit the indices 1 and 2.

16 | 2 Geometry of Physical Space where limh→0,a+h∈X φ(h) = 0. The mapping La is called the total derivative of f at point a, and since it is the unique mapping that satisfies (2.11), the following notation can be used: f 󸀠 (a) := La . The value of f 󸀠 (a) at an arbitrary vector is defined as follows: ∀h ∈ V1 : f 󸀠 (a)[h] =

f(a + th) − f(a) , X a;h ∋t→0 t lim

where X a;h = {t ∈ ℝ | t ≠ 0, a + th ∈ X}. Such a formula establishes the uniqueness of f 󸀠 (a). Since ℝ, V, V ∗ and, generally, T (k,l)(V) are affine spaces, the general notion of the total derivative is applied directly to them.

2.1.6 Integration Let D be generated by the Cartesian coordinate system (o, (i k )m k=1 ). An elementary volume in the Euclidean space E is represented by m-brick Im := D−1 (I m ), where I m = [a1 , b 1 ] × ⋅ ⋅ ⋅ × [a m , b m ] and [a i , b i ] ⊂ ℝ, i = 1, . . . , m. If f : E → ℝ is a function, then we define the integral over Im as b1

bm

∫ f := ∫ ̃f = ∫ dx1 ⋅ ⋅ ⋅ ∫ dx m ̃f , Im

Im

a1

am

where repeated integrals are treated as Riemannian or Lebesque integrals. Let T : E → T (k,l) (V) be a (k, l)-type tensor field. Define the integral over Im similarly to the scalar case: b1

bm

T = ∫ dx1 ⋅ ⋅ ⋅ ∫ dx m ̃ T. ∫ T := ∫ ̃ Im

Im

a1

am

i m In a basis (e i )m i=1 and its dual, (e )i=1 , one has

T=T

i 1 ...i k

j1 ...j l e i 1

⊗ ⋅ ⋅ ⋅ ⊗ e i k ⊗ e j1 ⊗ ⋅ ⋅ ⋅ ⊗ e j l ,

i ...i

where T 1 k j1 ...j l : E → ℝ are component functions. Thus, the integral can be represented in the form b1

bm

̃ ∫ T = e i1 ⊗ ⋅ ⋅ ⋅ ⊗ e i k ⊗ e j1 ⊗ ⋅ ⋅ ⋅ ⊗ e j l ∫ dx1 ⋅ ⋅ ⋅ ∫ dx m T Im

a1

i 1 ...i k j1 ...j l

.

am

The integral of a tensor field was reduced to the integral of scalar mappings (it is assumed that all transformations are well defined).

2.1 Euclidean Affine Space | 17

2.1.7 Symmetrization and Antisymmetrization Suppose that A is a finite-dimensional affine space with a distance function generated by the norm on the translation space V via (2.6). Consider a smooth vector field, u, and a covector field, ν, defined on some open set O ⊂ A. If (e i ) is some basis for V and (e i ) is its dual basis for V ∗ , then Du = ∂ k u s e s ⊗ e k ,

Dν = ∂ k ν s e s ⊗ e k .

In the linear approach adopted in convectional continuum mechanics, the symmetric and antisymmetric parts of the gradient of a vector field play the role of infinitesimal strains and vorticity tensors. Here, we do not presuppose that any Euclidean structure is defined on A. Thus, we have to introduce an operation similar to transposition without reference to any inner product. To this end, we define the operation (⋅)† , that is, the transposition of the elements in dyads: (a ⊗ b)† := b ⊗ a. Such definition coincides with the definition for transposition of the second-rank tensors in the Cartesian description of Euclidean space. Performing this operation (⋅)† for ν, we observe that expressions 1 1 (Dν + Dν† ) , (Dν − Dν† ) 2 2 are well defined, but at the same time the expressions 1 (Du + Du † ) , 2

1 (Du − Du † ) , 2

for u are ill defined, since e s ⊗ e k and e k ⊗ e s belong to different tensor spaces. The situation can be improved by introducing an inner product (⋅) on A that turns it into Euclidean affine space E. Then, the transposed tensor L T ∈ Lin(V; V) to a tensor L ∈ Lin(V; V) can be defined as¹³ ∀u, v ∈ V : v ⋅ L[u] = u ⋅ L T [v] . The symmetrization and antisymmetrization of u are well defined and can be written using the notion of the transpose: 1 (Du + Du T ) , 2

1 (Du − Du T ) . 2

2.1.8 Divergence In order to achieve a complete analytical toolkit for field analysis, one has to supplement the above definitions with divergence and curl operators. Let O ⊂ E be an open set.

13 The general definition of transposed tensor is given in Chapter 12, Section 12.5.4 by formula (12.3).

18 | 2 Geometry of Physical Space Divergence of the vector field. If a vector field u : O → V is differentiable at a ∈ O, then the divergence of u at a is the scalar div u(a) := tr[u 󸀠 (a)] .

(2.12)

With respect to the orthonormal basis (i k )sk=1 , one has div u(a) = entiable at O, then one can define the divergence field div u : O → ℝ ,

∂u i (a) . ∂x i

If u is differ-

x 󳨃→ div u(x) .

Gauss’ theorem gives the following geometrical interpretation for div u(a) in case m = 3: 1 div u(a) = lim ∫ u ⋅ n dS . δ→0 Vol(V δ ) ∂V δ

Here, V δ is an open ball with a center at a and a radius δ; Vol(V δ ) is its volume and n is an outward unit normal. Divergence of the tensor field. Suppose that a (1, 1)-tensor field T : O → T (1,1) (V) is differentiable at a ∈ O. Here, T (1,1) (V) = Lin(V; V). Then, the field T T : O → T (1,1) (V) is also differentiable at a ∈ O. The divergence of T at a is a unique vector div T(a) that is defined by the following relation: u ⋅ div T(a) = div{T T (a)[u]} , for every vector u ∈ V. If T = T kl i k ⊗ i l , then div T(a) = at O, then one has the divergence field div T : O → V ,

(2.13) ∂T kl ∂x l

i k . If T is differentiable

x 󳨃→ div T(x) .

Suppose that f : O → ℝ, u : O → V, L : O → Lin(V; V) are differentiable scalar, vector, and tensor fields, respectively. The following identities hold: div(f u) = u ⋅ grad f + f div u , T

󸀠

div(L[u]) = u ⋅ div L + tr(Lu ) .

(2.14) (2.15)

2.1.9 Curl Again, let O ⊂ E be an open set. Curl of the vector field. A curl of a vector field at some point is a vector that is associated with the antisymmetric part of its gradient. More precisely, suppose that a vector field u : O → V is differentiable at a ∈ O. The curl of u at a is the vector curl u(a), which is defined by the following relation: (Du(a) − Du T (a))v = (curl u(a)) × v ,

(2.16)

2.2 Curvilinear Coordinates

| 19

for every constant vector v. If u = u k i k , then direct calculations with (2.3) show that curl u(a) = e skq δ ql ∂ k u l (a)i s . Stokes’ theorem gives the following geometric interpretation for curl u(a) in the case m = 3: n ⋅ curl u(a) = lim

δ→0

1 ∫ u ⋅ τ dA , Area(D δ ) ∂D δ

where D δ is a two-dimensional disc centered at a with unit normal n; Area(D δ ) is its area and τ is the unit tangent vector to ∂D δ . If u is differentiable everywhere at O, then we define the field curl u : O → V ,

x 󳨃→ curl u(x) .

Curl of the tensor field. Let a (1, 1)-tensor field T : O → T (1,1) (V) be differentiable at a ∈ O. Then, the curl of T at a is a unique (1, 1)-tensor curl T(a) with the property: (curl T(a))u = curl{T T (a)[u]} ,

(2.17)

for every vector u ∈ V. If T = T sk i s ⊗ i k , then curl T(a) = e srq δ kl ∂ r T lq (a)i s ⊗ i k . If T is differentiable everywhere at O, then we define the field curl T : O → Lin(V; V) ,

x 󳨃→ curl T(x) .

2.2 Curvilinear Coordinates Curvilinear coordinates are defined as a local reparametrization of the Cartesian chart on the open subset of affine-Euclidean space. In fact, a new coordinate chart is added to the atlas Aaffine . This chart is related with the Cartesian one by smooth transition mappings. As a result, the points of E are identified directly with a tuple of coordinates. Thus, formally, the translation vectors are no longer required, and the Euclidean properties of E, declared by H. Weyl axioms, result from specific properties of the metric. Certainly, it is worth remembering that these specific properties of the metric are a consequence of the fact that it is calculated by reparametrization of the Cartesian (unit) metric, which, in turn, is obtained by H. Weyl axiomatic consideration. It should be noted that such reparametrization results in a structure that formally defines the Riemannian smooth manifold, although, of course, all non-Euclidean features are fictional, because they are the consequence of the change of variables only.

2.2.1 Reparametrization Let (E, V, (⋅), ψ) be an m-dimensional Euclidean affine space. Suppose that U ⊂ E is an open set. Its points can be uniquely described by the affine coordinates (x i )m i=1 in

20 | 2 Geometry of Physical Space m the coordinate system (o, (e i )m i=1 ). Let us assume that there is an open set G ⊂ ℝ and 1 a C -diffeomorphism¹⁴ ξ : G → U. Then, each point x ∈ U can be uniquely described by m-tuple (q1 , . . . , q m ) = ξ −1 (x) .

Numbers q i are called curvilinear coordinates of the point x. The ordered pair (U, ξ −1 ) represents a curvilinear chart on E. The whole set E may be covered by a number of charts (U k , ξ k−1 ), ξ k : G k → U k , which together constitute an atlas Acurvilin = {(U k , ξ k−1 )}Nk=1 . The coordinate representation of each map ξ k , k = 1, . . . , N, is the following: ξ̃k = D ∘ ξ k : G → ℝm ,

ξ̃k : (q1 , . . . , q m ) 󳨃→ (x1 , . . . , x m ) . i

∂x For each point x ∈ G, the Jacobian J = det( ∂q j ) is not equal to zero. A point x ∈ E is represented in curvilinear coordinates as

x = o + ̃x ik (q1 , . . . , q m )e i ,

̃ (̃x1k , . . . , ̃x m k ) = ξk ,

where

for some k = 1, . . . , N. Each field (scalar or, generally, tensor) u : O → T (r,l)(V), where ̃ q;k = u ∘ ξ k . In particular, for the O ⊂ E is open, can be parameterized locally as u position vector field (2.1), one has ̃ k = p|U k ∘ ξ k : G k → V . p In terms of the functions ̃x sk , the value of such a field can be represented as follows: ̃ k (q1 , . . . , q m ) = ̃x sk (q1 , . . . , q m )e s . p

(2.18)

̃ k is illustrated on the diagram The relation between ξ k , p, and p ℝm

⊃

ξk

Gk

p̃

Uk

⊂

E

p

k

V Table 2.1 contains some familiar examples of curvilinear coordinates in three-dimensional space.

14 That is, ξ is a bijection of class C 1 with its inverse.

2.2 Curvilinear Coordinates

| 21

Tab. 2.1: Curvilinear coordinates, m = 3. Coordinates

̃ xs

Cylindrical

(r, φ, z)

̃x 1 (r, φ, z) = r cos φ, ̃x 2 (r, φ, z) = r sin φ, ̃x 3 (r, φ, z) = z

Spherical

(r, θ, φ)

̃x 1 (r, θ, φ) = r sin θ cos φ, ̃x 2 (r, θ, φ) = r sin θ sin φ, ̃x 3 (r, θ, φ) = r cos θ

2.2.2 Coordinate Curves We will work with a single chart (U, ξ −1 ) ∈ Acurvilin . Suppose that x ∈ U has curvilinear coordinates ξ −1 (x) = (q10 , . . . , q0m ). The family {χ x;i }m i=1 of m curves that pass through x can be defined as follows: χ x;i : 𝕀i → E ,

χ x;i = ξ (q10 , . . . , q0i−1 , ⋅, q0i+1 , . . . , q0m ) .

Here, 𝕀i = {y ∈ ℝ | (q10 , . . . , q0i−1 , y, q0i+1 , . . . , q0m ) ∈ B x } is an interval; B x ⊂ G is an open ball centered at (q10 , . . . , q0m ). A curve with index i is obtained by fixing all arguments of ξ , except for the argument with number i. Carrying out such a procedure at each point x ∈ U results in a net of curves {χ x;i }i=1,...,m,x∈U . We refer to each such curve as a coordinate curve.

2.2.3 Local Basis Definition. The mapping (2.18) or, similarly, coordinate curves, generate m vectors at x ∈ U: ε s |x :=

̃ (q10 , . . . , q0m ) ∂p ∂q s

=

∂χ x;s (q10 , . . . , q0m ) ∂q s

,

i

s = 1, . . . , m .

∂̃x Since det ( ∂q j ) ≠ 0, these vectors are linearly independent and form a basis of V. Thus, one has the fields ε s : U → V, s = 1, . . . , m, the values of which form a basis of V. We refer to the collection (ε s )m s=1 as a local basis. m The transformation from the constant basis (e s )m s=1 into the local basis (ε s )s=1 is i provided by a field [Ω j ] : U → GL(m; ℝ) of non-degenerate m × m matrices. At a point m x ∈ U, the transformation (e s )m s=1 󳨃→ (ε s | x )s=1 is defined by 󵄨 ∂ x̃ i 󵄨󵄨󵄨 󵄨󵄨 . ε s |x = Ω i s |x e i , where Ω i s |x = ∂q s 󵄨󵄨󵄨ξ −1 (x) m The inverse transformation, (ε s |x )m s=1 󳨃→ (e s )s=1 , is given by

e s = ℧i s |x ε i |x .

22 | 2 Geometry of Physical Space Here, [℧i s ] = [Ω i j ]−1 , or, pointwise, at x ∈ U, ℧i s |x Ω s j |x = δ ij , i, j = 1, . . . , m. Thus, at a point x ∈ U, 󵄨 ∂ q̃ i 󵄨󵄨󵄨 󵄨󵄨 , ℧i s |x = s ∂x 󵄨󵄨󵄨x where q̃ i : (x1 , . . . , x m ) 󳨃→ (q1 , . . . , q m ) is the coordinate representation of ξ −1 . Consider the relation between ε i |x and ε i |y , where x, y ∈ U. Using inverse transformation one can write ℧i s |x ε i |x = ℧i s |y ε i |y . Multiplying both sides by Ω s j |y and summing over s gives j ε i |y = Ω s i |y ℧ s |x ε j |x , i = 1, . . . , m . (2.19) s ∗ The dual basis (ε s |x )m s=1 consists of covector fields ε : U → V , which are related with the local basis as follows:

x ∈ U : ⟨ε i |x , ε j |x ⟩ = δ ij , If

(e s )m s=1

i, j = 1, . . . , m .

is the dual basis that corresponds to (e s )m s=1 , then one has ε i |x = ℧i s |x e s .

Representations in a local basis. Consider representations of geometric objects in terms of the local basis. For a point x ∈ U, one obtains j

x = o + x̃ ik (q1 , . . . , q m )℧ i |x ε j |x . If v = v i e i is a vector from V, then for each x ∈ U one arrives at j

v = v i ℧ i |x ε j |x = w j |x ε j |x , j

where w j |x = v i ℧ i |x . Note that the numbers v i transform into numbers w j |x , which change from point to point, i.e., numerical components of a vector are represented by functions on U! Let v : U → V be a vector field. It is represented by the decomposition v = v i e i , where v i : U → ℝ are functions. Representing it in the local basis: ∀x ∈ U :

v|x = w j |x ε j |x ,

where

j

w j |x = v i |x ℧ i |x .

2.2.4 From Pointwise to the Whole In the previous arguments we used pointwise notation, where one writes relations between values of the tensor fields. Such a notation is quite cumbersome, since one needs to write arguments in every slot. Further, we use pointwise notation only if there is a danger of confusion. If f is some function and u is a vector field, then f u means m the vector field x 󳨃→ f|x u|x . The relations between bases (e s )m s=1 and (ε s )s=1 are written as ε s = Ω i s e i , e s = ℧i s ε i . The decomposition u x = u i |x ε i |x of a vector field u is represented by u = u i ε i .

2.2 Curvilinear Coordinates

|

23

2.2.5 Metric Tensors The metric tensor on V. The scalar product (⋅) on V generates a constant tensor field g : E → V ∗ ⊗ V ∗ , referred to as metric tensor: ∀u, v ∈ V :

g(u, v) := u ⋅ v .

i j In a constant basis (e i )m i=1 , the metric tensor can be decomposed as g = g ij e ⊗ e , m where g ij := e i ⋅ e j . If, in addition, (e i )i=1 is orthonormal, then g ij = δ ij . Hence, the components of g, namely g ij , are constant functions in rectilinear coordinates. Meanwhile, if one uses curvilinear coordinates, the components of g in local basis (ε i )m i=1 become non-constants: g ij = ε i ⋅ ε j ≠ const.

Metric coefficients. Components δ ij of metric tensors in Cartesian coordinates and components g ij of metric tensors in curvilinear coordinates are related as follows. Since, 󵄨 ∂ ̃x k 󵄨󵄨󵄨 󵄨󵄨 ik , ε s |x = s ∂q 󵄨󵄨󵄨ξ −1 (x) we obtain the explicit expression for calculating the metric coefficients g kl = ε k ⋅ ε l : g kl |x = ε k |x ⋅ ε l |x = δ ij

󵄨 󵄨 ∂ ̃x i 󵄨󵄨󵄨 ∂ ̃x j 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 . ∂q k 󵄨󵄨󵄨ξ −1 (x) ∂q l 󵄨󵄨󵄨ξ −1 (x)

Geometrical measurements using metric tensors. Let u : U → V be a vector field. At each point x ∈ U one can calculate the norm ‖u x ‖ of its value at that point. Thus, the function x 󳨃→ ‖u x ‖ is defined. This function can be expressed as ‖u‖ = √g(u, u), or, in curvilinear coordinates, ‖u‖ = √g ij u i u j , where u = u i ε i . For vector fields u : U → V, v : U → V, one can obtain their scalar product, i.e., the mapping x 󳨃→ u x ⋅ v x , using the metric tensor: u ⋅ v = g ij u i v j , where u = u i ε i , v = v i ε i . Finally, angles between u, v are measured pointwise. In terms of the cosine map, g ij u i v j cos(u, v) = . √g ij u i u j √g ij v i v j Metric tensors on V ∗ . The metric tensor g and the musical isomorphisms induce the ∗ associated metric tensor g on V ∗ as follows: ∗

g := g ♯ = g ij ε i ⊗ ε j . The matrices of components of these tensors are mutually inverse: [g ij ] = [g ij ]−1 .

24 | 2 Geometry of Physical Space

2.2.6 Parallel Transport Consider the cosine map cos (2.7). Let u be a vector field on U, u x = u i |x ε i |x , where x ∈ U. Then, for x, y ∈ U, cos(u x , u y ) =

u i |x u j |y Ξ i,j (x, y) √g kl |x u k |x u l |x √g kl |y u k |y u l |y

,

where Ξ i,j (x, y) = ε i |x ⋅ε j |y = e k ⋅e l Ω k i |x Ω l j |y . Thus, the vector field u is weakly parallel transported along a curve χ : [a, b] → U if 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 u i |χ(a) u j |χ(x) Ξ i,j (χ(a), χ(x)) 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 1 . ∀x ∈ [a, b] : 󵄨󵄨󵄨 󵄨󵄨 √g | u k | u l | √g | u k | u l | 󵄨󵄨󵄨 󵄨󵄨 kl χ(a) χ(a) χ(a) kl χ(x) χ(x) χ(x) 󵄨󵄨 The equivalent representation, u(x) = f(x)u(χ(a)), is represented in coordinates as u i |x = f(x)u j |a Ω s j |y ℧i s |x , where the relation (2.19) was used.

2.2.7 Total Derivatives The case of the vector field. Suppose that u : U → V is a differentiable vector field. i One has Du = ∂ j u i e i ⊗ e j . Consider the similar field in the local basis (ε s )m s=1 : u = u ε i . Applying the product rule gives¹⁵ Du = D (u i ε i ) = ε i ⊗ D(u i ) + u i D(ε i ) . Remark 2.4. Let us obtain this formula using the definition of the total derivative. Suppose that a ∈ U. Choose some h ∈ V such that a + h ∈ U. Then, u(a + h) − u(a) = u i (a + h)ε i (a + h) − u i (a)ε i (a) = {u i (a) + Du i (a)[h] + o(‖h‖)}{ε i (a) + Dε i (a)[h] + o(‖h‖)} − u i (a)ε i (a) = u i (a)Dε i (a)[h] + ε i (a)Du i (a)[h] + o(‖h‖) . Since total derivative is unique, one obtains Du(a)[h] = u i (a)Dε i (a)[h] + ε i (a)Du i (a)[h] , and thus, Du(a) = u i (a)Dε i (a) + ε i (a) ⊗ Du i (a) . 15 The mapping (λ, x) 󳨃→ λx is bilinear and one applies the product rule to it.

2.2 Curvilinear Coordinates

|

25

We will consider the total derivative Du, in which u i and ε i are thought of as functions of (q i ). Applying the chain rule gives ∂u i j ∂u i k e = ε , ∂x j ∂q k ∂ε i ∂ε i ⊗ ej = ⊗ εk . D(ε i ) = ∂x j ∂q k

D(u i ) =

The vector field

∂εi ∂q k

can be represented as ∂ε i = Γ lik ε l . ∂q k

(2.20)

The functions Γ lik : U → ℝ are referred to as the Christoffel symbols (of the second kind¹⁶). Finally, Du has the form Du = (

∂u i + u j Γ ijk ) ε i ⊗ ε k . ∂q k

Christoffel symbols. Consider some properties of functions Γ lik . From now on we suppose that ξ is a C2 -diffeomorphism. Since ̃ ̃ ∂ε i ∂2 p ∂2 p ∂ε k = = = , ∂q i ∂q k ∂q k ∂q i ∂q i ∂q k one obtains that Christoffel symbols are symmetric in the lower two indices: Γ lik = Γ lki ,

i, k, l = 1, . . . , m .

Next, one can obtain the relation between Christoffel symbols and functions g ij = ε i ⋅ εj . Differentiating the relation g ij = ε i ⋅ ε j with respect to q k one arrives at ∂ k g ij = ε j ⋅ ∂ k ε i + ε i ⋅ ∂ k ε j . Using the formula (2.20) gives ∂ k g ij = g lj Γ lik + g il Γ ljk .

16 There are two types of Christoffel symbols. The first one is represented by functions Γ lik = g ls Γ sik , which are called Christoffel symbols of the first kind. The second type is represented by functions Γ lik , which are called Christoffel symbols of the second kind. We deal only with the latter kind of Christoffel symbols. In this regard, we further omit the words “second kind”.

26 | 2 Geometry of Physical Space

By a cyclic permutation we obtain the expressions ∂ k g ij = g lj Γ lik + g il Γ ljk , ∂ i g jk = g lk Γ lji + g jl Γ lki , ∂ j g ki = g li Γ lkj + g kl Γ lij . Summing the latter two equations and subtracting the former gives 2Γ sij g sk = ∂ i g kj + ∂ j g ik − ∂ k g ij , where the symmetry of g ij and Γ lik was taken into account. Finally, one arrives at Γ lij =

g lk (∂ i g kj + ∂ j g ik − ∂ k g ij ) . 2

(2.21)

The Voss–Weyl formula. The Voss–Weyl formula ∂ k √g = √g Γ llk ,

g = det[g ij ] ,

(2.22)

is used in the following for the coordinate representation of divergence. In order to derive it, put i = l in (2.21): Γ llk =

g lr 1 ∂ k g lr = ∂ k ln g . 2 2

To prove that g lr ∂ k g lr is equal to ∂ k ln g, consider the function f : GL(m; ℝ) → ℝ, which acts as follows: f(A) = ln | det A| . Let δA be an increment of A. Then, for sufficiently small δA, f(A + δA) − f(A) = ln(det (I + A −1 δA)) . Using the expansion det (I + α) = 1 + tr α + O(α 2 ), we obtain f(A + δA) − f(A) ≈ ln(1 + tr (A −1 δA)) . This means that δf = tr (A−1 δA). Applying this result to A = [g ij ], we arrive at δ ln g = g ij δg ij , and this implies the equality g lr ∂ k g lr = ∂ k ln g. Note that ∂ k ln g =

2 ∂k g = ∂ k √g . g √g

This observation leads to the desired Voss–Weyl formula (2.22). Table 2.2 contains non-zero Christoffel symbols and the matrices of metric for cylindrical and spherical coordinates.

2.2 Curvilinear Coordinates

|

27

Tab. 2.2: Christoffel symbols and metric. Coordinates

Metric

Christoffel symbols

(r, φ, z)

diag {1, r 2 , 1}

Γ 122 = −r, Γ 221 = Γ 212 =

(r, θ, φ)

diag {1,

1 r

Γ 122 = −r, Γ 133 = −r sin2 θ, r2

2

2

, r sin θ}

Γ 221 = Γ 212 = Γ 313 = Γ 331 =

1 , r

Γ 233 = − cos θ sin θ, Γ 323 = Γ 332 = cot θ

Dual basis. Consider the dual basis (ε i ). Since each element of the dual basis is a mapping with values in V ∗ , one has that ∂ k ε i similarly takes values in V∗ . Let C i jk : U → ℝ be smooth mappings such that ∂ k ε i = C i jk ε j . In fact, C i jk = −Γ ijk . To prove this, it is sufficient to differentiate the equality ⟨ε i , ε i ⟩ = δ ij with respect to q k . Indeed, 0 = ⟨∂ k ε i , ε j ⟩ + ⟨ε i , ∂ k ε j ⟩ = C i jk + Γ ijk . Thus, ∂ε i = −Γ ijk ε j . ∂q k

(2.23)

(1, 1)-tensor field. Suppose that T : U → Lin(V; V) is a differentiable tensor field. One has the decomposition T = T ij ε i ⊗ ε j . Let a ∈ U be a point and let h ∈ V be a vector such that a + h ∈ U. One has DT(a)[h] = ε i (a) ⊗ ε j (a)DT i j (a)[h] + T ij (a)Dε i (a)[h] ⊗ ε j (a) + T i j (a)ε i (a) ⊗ Dε j (a)[h] . Remark 2.5. This formula can be obtained in a similar manner as for the vector field. That is, one needs to consider the increment T(a + h) − T(a): T(a + h) − T(a) = T ij (a + h)ε i (a + h) ⊗ ε j (a + h) − T ij (a)ε i (a) ⊗ ε j (a) = = ε i (a) ⊗ ε j (a)DT i j (a)[h] + T ij (a)Dε i (a)[h] ⊗ ε j (a)+ + T ij (a)ε i (a) ⊗ Dε j (a)[h] + o(‖h‖) . The latter equality and the uniqueness of total derivative give the desired formula. Let h k = ⟨ε k , h⟩, k = 1, . . . , m. Since DT i j =

∂T ij ∂q

εk , k

D(ε i ) =

∂ε i ⊗ εk , ∂q k

Dε j =

∂ε j ⊗ εk , ∂q k

one obtains (here we omit the argument a) i { ∂T j ∂ε i ∂ε j } DT[h] = h k { k ε i ⊗ ε j + T ij k ⊗ ε j + T ij ε i ⊗ . ∂q ∂q ∂q k } { }

28 | 2 Geometry of Physical Space

Taking into account the relations (2.20) and (2.23), one obtains i { ∂T j } DT[h] = h k { k + T lj Γ ilk − T il Γ ljk } ε i ⊗ ε j . ∂q { }

Thus, finally, i } { ∂T j DT = { k + T lj Γ ilk − T il Γ ljk } ε i ⊗ ε j ⊗ ε k . ∂q } {

2.2.8 Integration Suppose that D is generated by the Cartesian coordinate system (o, (i k )m k=1 ) and Im is an m-brick (see Section 2.1.6). Suppose that Im ⊂ U and f : U → ℝ is a function, for which the integral ∫I f exists. Using the change of variables theorem one arrives at m

∫f = Im

∫

Jf ∘ ξ ,

ξ −1 (I m )

i

∂x where J = det( ∂q j ).

2.2.9 The Physical Basis Suppose that the local basis (ε s )m s=1 is orthogonal: g ij = ε i ⋅ ε j = 0 , if i ≠ j. Then, one can introduce the collection of fields (ε ⟨s⟩ )m s=1 , ε⟨s⟩ :=

1 εs , ‖ε s ‖

where

‖ε s ‖ = √g ss .

The values of the obtained fields (ε⟨s⟩ )m s=1 form a basis of V at every point of U. In general, these fields are not tangent to coordinate curves. The basis (ε ⟨s⟩ )m s=1 is called the physical basis.

2.3 Nabla Operator Formalism in 3D-Space The symbolic vector ∇ is heavily used in continuum mechanics [45]. It is alternative way to obtain differential expressions for scalar, vector, and tensor fields. Let E be a three-dimensional Euclidean affine space.

2.3 Nabla Operator Formalism in 3D-Space | 29

2.3.1 Definitions of Nabla Nabla in Cartesian coordinates. Introduce a symbolic vector ∇, which is represented in Cartesian coordinates as follows [46]: ∇ = ik

∂ , ∂x k

where (i k )3k=1 is the dual vector basis that coincides with the orthonormal basis (i k )3k=1 , i.e., i k = i k , for all k = 1, 2, 3. From this point of view, applied to the ordered pair (V, V ∗ ), vectors i k may be regarded as covectors i k , i.e., elements of the dual space V ∗ . Nabla in curvilinear coordinates. Suppose that some curvilinear coordinates (q i )3i=1 are chosen on some open set U ⊂ E. If (ε s )3s=1 is the local basis, then let (ε s )3s=1 denote the dual vector basis, ε i ⋅ ε j = δ ij , i, j = 1, 2, 3 . The chain rule implies that nabla operator would take the form ∇ = εk

∂ . ∂q k

Nabla in the physical basis. If the basis (ε s )3s=1 is orthogonal, then in the physical basis (ε ⟨s⟩ )3s=1 one has 3 ε⟨s⟩ ∂ . ∇= ∑ s √g ss ∂q s=1 Table 2.3 contains expressions for ∇ in the case of cylindrical and spherical coordinates. Tab. 2.3: Nabla operator in physical basis. Coordinates

∇

(r, φ, z)

∇ = ε⟨r⟩

(r, θ, φ)

ε ⟨φ⟩ ∂ + ∂r r ε ⟨θ⟩ ∂ ∇ = ε⟨r⟩ + ∂r r

∂ ∂ + ε⟨z⟩ ∂φ ∂z ε⟨φ⟩ ∂ ∂ + ∂θ r sin θ ∂φ

2.3.2 Operations in Euclidean Space The application of ∇ presupposes the identification of vectors and covectors (due to musical isomorphisms). Let (e i )3i=1 be a basis for V and (e i )3i=1 be the corresponding

30 | 2 Geometry of Physical Space dual vector basis. For a vector u ∈ V, one has the decompositions u = ui ei = ui ei . Introduce the following particular definition of tensor product (this definition holds only in Euclidean spaces). If u, v ∈ V, then [47] u ⊗ v: V → V ,

u ⊗ v(w) := (v ⋅ w)u .

It is linear map, an element of Lin(V; V). Left and right “dot” actions of u ⊗ v on a vector w are defined via (u ⊗ v) ⋅ w := (v ⋅ w)u ,

w ⋅ (u ⊗ v) := (u ⋅ w)v .

The “cross”-action of the dyad u ⊗ v on a vector w is defined as follows: (u ⊗ v) × w := u ⊗ (v × w) ,

w × (u ⊗ v) := (w × u) ⊗ v .

In Euclidean space, second-order tensors of all four types are identified. Let (e i )3i=1 be a basis for V and (e i )3i=1 be the corresponding dual vector basis. Then each of (e i ⊗ e j ) ,

(e i ⊗ e j ) ,

(e i ⊗ e j ) ,

(e i ⊗ e j ) ,

form a basis for Lin(V; V). Any element T ∈ Lin(V; V) has the decompositions j

T = T ij e i ⊗ e j = T ij e i ⊗ e j = T i e i ⊗ e j = T ij e i ⊗ e j . If (e i )3i=1 is a basis for V and (e i )3i=1 is the corresponding dual vector basis, then introduce quantities ϵ ijk , ϵ ijk , such that e i × e j = ϵ ijk e k ,

e i × e j = ϵ ijk e k .

In classical books it is shown that [45] ϵ ijk = √g e ijk ,

ϵ ijk =

1 ijk e . √g

(2.24)

Recall that g = det[g ij ], g ij = e i ⋅ e j .

2.3.3 Nabla and Mappings Nabla and scalar functions. Consider a differentiable function f : U → ℝ. One has ∇f = i k and in this regard, ∇f = grad f (2.9).

∂f , ∂x k

2.3 Nabla Operator Formalism in 3D-Space | 31

Nabla and vector fields. One can form the following combinations with ∇ and operations ⊗, (⋅), ×: ∇⊗ = i k ⊗

∂ , ∂x k

∇⋅ = i k ⋅

∂ , ∂x k

∇× = i k ×

∂ . ∂x k

(2.25)

Consider a differentiable vector field u : U → V. Applying ∇⊗ on u = u s i s gives ∇⊗u =

∂u s k i ⊗ is . ∂x k

Let a ∈ U be a point. Then, for a vector h ∈ V, such that a + h ∈ U, one has two possible notations u(a + h) = u(a) + Du(a)[h] + o(‖h‖)

as

h→0,

u(a + h) = u(a) + h ⋅ ∇ ⊗ u|x=a + o(‖h‖)

as

h→0.

Thus, Du(a) = {∇ ⊗ u|x=a }T . Applying ∇⋅ on u = u s i s gives the divergence of u (2.12): div u = ∇ ⋅ u . Finally, consider ∇×. This gives the curl of u (2.16): curl u = ∇ × u . Nabla and (1, 1)-tensor fields. Again, one can consider combinations (2.25). Consider a differentiable tensor field T : U → Lin(V; V). At a point a ∈ U, one has T(a + h) = T(a) + DT(a)[h] + o(‖h‖)

as

h→0,

T(a + h) = T(a) + h ⋅ ∇T| x=a + o(‖h‖)

as

h→0

Like for vector fields one has the equality div T = ∇ ⋅ T , for divergence (2.13) and the equality curl T = ∇ × T , for curl (2.17). Remark 2.6. The curl operation is used especially in the linear theory of elasticity [47]. In this book one will meet this operation in arguments on incompatibility, where, among general considerations, the linear approach is analyzed. In the rest of the book, we use the generalization of the curl, which is based on the exterior differential.

32 | 2 Geometry of Physical Space

2.3.4 Coordinate Representations for Divergence In Section 2.1.8, we obtained representations for divergences of the smooth vector u and the tensor T fields in Cartesian coordinates (x i ): div u = ∂ x i u i ,

div T = ∂ x l T kl i k .

Operator ∇ gives a convenient way to calculate divergences in curvilinear coordinates (q k ). The vector field. Let u = u i ε i be a smooth vector field represented in curvilinear coordinates. Then, div u = ∇ ⋅ u, and div u = ε k ⋅

∂ (u i ε i ) = ε k ⋅ (∂ k u i + u j Γ ijk ) ε i , ∂q k

from where it follows that div u = ∂ i u i + u j Γ iji , since ε k ⋅ ε i = δ ki . Using the Voss–Weyl formula (2.22), ∂ j √g = √g Γ iij , one obtains div u = ∂ i u i + (1/√g)u i ∂ i √g, and, finally, div u =

1 ∂ (u i √g) . √g ∂q i

(2.26)

Suppose that (ε i ) is orthogonal. Consider div u in the physical basis (ε s ). Let u = u ⟨i⟩ ε⟨i⟩ . Here, u ⟨i⟩ = u i √g ii . One can obtain the expression for div u by direct differentiation or by changing u i in (2.26). In all cases, one obtains div u =

g 1 3 ∂ ∑ (u ⟨i⟩ √ ) . i g √g i=1 ∂q ii

(2.27)

The tensor field. Let T = T ij ε i ⊗ ε j be a smooth tensor field. One has div T = ∂ k T kj ε j + T kj (ε i ⋅ ∂ i ε k ) ε j + T kj ∂ k ε j .

(2.28)

Since ε i ⋅ ∂ i ε k = Γ iik , one can use the Voss–Weyl formula (2.22). Thus, from (2.28), one obtains ∂ k √g ε j + T kj ∂ k ε i . div T = ∂ k T kj ε j + T kj √g Finally, one obtains div T =

1 ∂ {√g T kj ε j } . √g ∂q k

There is an alternative formula. Starting from (2.28), one arrives at j

div T = (∂ k T kj + T lj Γ klk + T kl Γ lk ) ε j .

2.4 Riemannian Space | 33

Using the Voss–Weyl formula (2.22) and the relation (2.21), one has div T = (∂ k T kj + T lj

∂ l √g T kl g js + (∂ l g sk + ∂ k g ls − ∂ s g lk )) ε j . 2 √g

Assume that (ε i ) is an orthogonal basis. Consider div T in the physical basis (ε s ). Since T ⟨kj⟩ = T kj √g kk √g jj , one has div T =

g 1 3 ∂ ∑ ε⟨j⟩ } . {T ⟨kj⟩ √ g kk √g k=1 ∂q k

2.3.5 Coordinate Representations for Curl The vector field. Let u = u i e i be a smooth vector field. Then, j

∇ × u = e k × e i (∂ k u i − u j Γ ik ) . Using (2.24), one obtains j

∇ × u = ϵ kil e l (∂ k u i − u j Γ ik ) . j

j

j

Consider the term ϵ kil Γ ik , where l and j are fixed. Since ϵ kil = −ϵ ikl and Γ ik = Γ ki , one arrives at j j j ϵ kil Γ ki = −ϵ ikl Γ ik = −ϵ kil Γ ki , j

and this implies ϵ kil Γ ki = 0. Finally, curl u = ϵ kil ∂ k u i e l . The tensor field. Consider the tensor field T = T ij e i ⊗ e j . One has ∇ × T = e k × (∂ k T ij e i ⊗ e j − T ij Γ ilk e l ⊗ e j + T ij e i ⊗ ∂ k e j ) = ϵ kil e l ∂ k (T ij e j ) − T ij ϵ kls Γ ilk e s ⊗ e j . Like in the case of vector field, one obtains ϵ kls Γ ilk = 0, and, thus one arrives at the final expression curl T = ϵ kil e l ∂ k (T ij e j ) .

2.4 Riemannian Space 2.4.1 General Definition From general point of view, the Riemannian space of a dimension m is a 5-tuple (P, S, D, g, μ) ,

34 | 2 Geometry of Physical Space where P is a set whose elements are referred to as spatial points, S is a topology on P, D is a smooth structure on P, g is a Riemannian metric on P, and μ is a volume form on P. The topological space (P, S) is required to be Hausdorff and second countable¹⁷. The triple (P, S, D) is a smooth m-dimensional manifold. The smooth structure D is introduced by a chosen equivalence class of smooth atlases and defines the maximal atlas Amax , i.e., the union of all atlases from this equivalence class¹⁸. Any chart (U, φ) ∈ Amax , where U ⊂ P is an open set, establishes local coordinates for all points from U. We use the following terminology: any chart from the maximal atlas is called a smooth chart. If two charts (U, φ), (V, ψ) ∈ Amax overlap, i.e., U ∩ V ≠ 0, then the transition map ψ ∘ φ−1 is a diffeomorphism between open subsets of ℝm . This map represents the transformation law between local coordinates. The Riemannian metric¹⁹ g represents a field of inner products on tangent spaces T p P. It is a (0, 2)-type tensor field on P, or, formally, a smooth section of the vector bundle T ∗ P ⊗ T ∗ P. For any point p ∈ P, one has the bilinear mapping gp , which is symmetric and positive definite. The volume form μ is a differential m-form, the element of Ω m (P) that satisfies the requirement: μ p ≠ 0, for each p ∈ P. Since P admits a volume form, it is orientable²⁰. We choose the orientation that is generated by μ. The structure of Riemannian space is more “flexible” than the structure of Euclidean space; one can define topology, local coordinates, and the metric directly. In Euclidean space, these features are induced by the structure of the translation vector space. Remark 2.7. Suppose that we are given an s-dimensional topological manifold M. Let # denote the number of C∞ manifolds that one can make out from M up to a diffeomorphism. Such a number depends on the dimension, s, of the manifold. Table 2.4 illustrates such dependence [48]. Tab. 2.4: Dimension and number of smooth structures. s

#

1, 2, 3 4 5, 6, 7

1 in general, uncountably many finite

17 The definitions of the Hausdorff and second countability properties are given in Chapter 13, Section 13.1.1. 18 The notion of smooth structure is introduced in Chapter 13, Section 13.1.2. 19 The definition is given in Chapter 13, Section 13.3.10. 20 See Chapter 13, Section 13.4.

2.4 Riemannian Space | 35

2.4.2 Smooth Mappings on P We repeat the definitions given in Section 2.1.3, but now they are adapted to the Riemannian geometry. Curves. A curve on P is a mapping χ : 𝕀 → P, where 𝕀 ⊂ ℝ is an interval. In a smooth chart (U, φ) a curve χ has the coordinate representation ̃χ = φ ∘ χ : 𝕀 → ℝm . The smoothness of the curve χ is understood in the sense of the smoothness of its coordinate representation ̃χ . Functions. A function on P is a mapping f : P → ℝ. Its coordinate representation relative to a smooth chart (U, φ) on P has the form: ̃f = f ∘ φ−1 : φ(U) → ℝ. The smoothness of f is defined by the smoothness of its coordinate representation ̃f ; we say that f is of class C∞ at a point p ∈ P, if for some smooth chart that contains p, the mapping ̃f is of C∞ . We deal with functions that are smooth everywhere in P. Such functions form the real algebra C∞ (P) under pointwise addition and multiplication²¹. If O ⊂ P is an open subset, then it is considered as an open submanifold of P, and one can introduce the algebra C∞ (O) of smooth functions on it. Vector fields. Unlike the Euclidean case, one has no translation vector space for the space P. A tangent vector²² at a point p ∈ P can be viewed as an equivalence class of curves that pass through p, or, equivalently, as a derivation on the space of germs at p, or, as a derivation of the algebra C ∞ (P) at p (see Chapter 13, Section 13.2.1). For a point p ∈ P, the tangent space to P at p is denoted by T p P. Suppose that f ∈ C∞ (P) and u p ∈ T p P is a vector. One can calculate the directional derivative at p: Lu p (f) =

󵄨󵄨 d 󵄨 , (f ∘ χ)(t)󵄨󵄨󵄨 󵄨󵄨 t=0 dt

where χ : 𝕀 → P is a smooth curve from u p (which is an equivalence class); χ(0) = p. Here, f ∘ χ : 𝕀 → ℝ. The first-order Taylor expansion holds for h ∈ 𝕀: (f ∘ χ)(h) = (f ∘ χ)(0) + hLu p (f) + o(h)

as

h →0.

In some smooth chart (U, φ) on P, the vector u p has the representation u p = u i ∂ i |p , where 󵄨󵄨 d 󵄨 . (φ ∘ χ)(t)󵄨󵄨󵄨 (u 1 , . . . , u m ) = 󵄨󵄨 t=0 dt The coordinate representation of f is the mapping ̃f : (x1 , . . . , x m ) 󳨃→ ̃f (x1 , . . . , x m ) . 21 The space C ∞ (P) is defined in Chapter 13, Section 13.1.3. The notion of algebra is defined in Chapter 12, Section 12.5.2. 22 Three equivalent definitions of tangent vector are given in Chapter 13, Section 13.2.1.

36 | 2 Geometry of Physical Space Thus, Lu p (f) is represented by Lu p (f) = u i

󵄨 ∂̃f 󵄨󵄨󵄨󵄨 . 󵄨 ∂x i 󵄨󵄨󵄨󵄨φ(p)

The collection of tangent spaces T p P at all points p ∈ P forms the tangent bundle TP = ⨆ p∈P T p P. The latter is a 2m-dimensional smooth manifold²³. A vector field is a global section u : P → TP of the tangent bundle or a local section u : O → TP of the tangent bundle²⁴. Here, O is an open subset of P. The term section means that for any p from P (or from O), one has u p ∈ T p P. Since a vector field is a mapping between manifolds, its smoothness is established as the smoothness of a mapping. If (U, φ) is a smooth chart on P, then one can introduce the coordinate frame²⁵ (∂ i )m i=1 , U ∋ p 󳨃→ ∂ i | p ∈ T p P. For a smooth vector field u, i one has u = u ∂ i . The set of all smooth vector fields on P is denoted by Vec(P). Suppose that u ∈ Vec(P). Since u p is a derivation on C∞ (P), for any f ∈ C∞ (P) one can define the mapping uf ∈ C∞ (P) as uf : p 󳨃→ u p (f). Another vector field v ∈ Vec(P) may act on such a function: v(uf) ∈ C∞ (P). Similarly, u(vf) ∈ C∞ (P). Both vector fields define the Lie bracket operation (13.20) [u, v]f = u(vf) − v(uf), where [u, v] ∈ Vec(P). The properties of the Lie bracket are given in Chapter 13, Section 13.3.5. Covector fields. A tangent covector at p ∈ P is an element of the cotangent space T ∗p P := (T p P)∗ . The set T ∗ P = ⨆ p∈P T ∗p P represents the cotangent bundle. As the tangent bundle, it is a 2m-dimensional smooth manifold²⁶. A covector field is a global section ν : P → T ∗ P of the cotangent bundle, or a local section ν : O → T ∗ P of the cotangent bundle. Here, O is an open subset of P. Any covector field is a mapping between manifolds, and its smoothness is established as the smoothness of a mapping. If (U, φ) is a smooth chart on P, then one can i ∗ introduce the coordinate coframe (dx i )m i=1 , U ∋ p 󳨃→ dx | p ∈ T p P. The coordinate frame and coframe are related via ⟨dx i , ∂ j ⟩ = δ ij ,

i, j = 1, . . . , m .

For a smooth covector field ν, one has ν = ν i dx i . The set of all smooth covector fields on P is denoted by CVec(P).

23 The smooth atlas is introduced for TP in Chapter 13, Section 13.3.2. 24 The notion of section is discussed in Chapter 13, Section 13.3.5. 25 Coordinate frames for different tensor fields are introduced in Chapter 13, Section 13.3.8. 26 Smooth atlas on T ∗ P is introduced in the analogous manner as for TP. See Chapter 13, Section 13.3.2.

2.4 Riemannian Space | 37

Tensor fields. A mixed (k, l)-tensor is an element of the vector bundle T (k,l)(TP) = ⨆ T (k,l)(T p P) . p∈P

As in the case of the vector and the covector field, a tensor field is a global or local section of this bundle. The coordinate frame is represented by the collection (∂ i1 ⊗ ⋅ ⋅ ⋅ ⊗ ∂ i k ⊗ dx j1 ⊗ ⋅ ⋅ ⋅ ⊗ dx j l )1≤i

1 ,...,i k ≤s

,1≤j1 ,...,j l ≤s

.

For a smooth mixed (k, l)-tensor T, one has T=T

i 1 ...i k

j1 ...j l ∂ i 1

⊗ ⋅ ⋅ ⋅ ⊗ ∂ i k ⊗ dx j1 ⊗ ⋅ ⋅ ⋅ ⊗ dx j l .

2.4.3 Geometric Measures and Geodesics Geometric measures. The machinery that allows one to determine lengths and angles is given by the Riemannian metric g. For a tangent vector v ∈ T p P, where p ∈ P is some point, the square of its length is defined by the formula ‖v‖2g = g(v, v) . If v is a velocity vector (13.18) of a curve χ, then ‖v‖g has the sense of the speed of this curve. If χ : ]a, b[ → P is a smooth curve on P, then its length, lg (χ), is, by definition, b

lg (χ) = ∫ √gχ(t) (χ 󸀠 (t), χ 󸀠 (t)) dt . a

Here, χ 󸀠 (t) is defined by (13.18). Let χ 1 : ]a, b[→ P, χ2 : ]c, d[→ P be two smooth curves that intersect at p = χ 1 (t1 ) = χ2 (t2 ). Denote the velocity vector of the curve χ i , i = 1, 2 by u i : t 󳨃→ u i (t) = χ 󸀠i (t). The angle ∠(χ 1 , χ 2 )p between the curves χ1 and χ 2 at the point p is the angle between their velocity vectors. That is, cos ∠(χ 1 , χ2 )p =

gp (u 1 (t1 ), u 2 (t2 )) . ‖u 1 (t1 )‖g ‖u 2 (t2 )‖g

Geodesics. For a smooth curve χ : ]a, b[ → P in a smooth chart, one has g = g ij dx i ⊗ dx j , χ(t) = (χ 1 (t), . . . , χ m (t)) and χ 󸀠 (t) = χ̇ i (t)∂ i |χ(t) (13.18). Then, one arrives at the coordinate expression for lg (χ): b

lg (χ) = ∫ L(χ i , χ̇ i ) dt , a

38 | 2 Geometry of Physical Space

where L(χ i , χ̇ i ) = √g ij (χ 1 , . . . , χ m )χ̇ i χ̇ j is the Lagrangian density. A geodesic is an extremal of the functional lg (χ): δlg (χ) = 0. The system of the Euler–Lagrange equations has the form ∂L d ∂L − ) = 0 , k = 1, . . . , m . ( ∂χ k dt ∂ χ̇ k Derivation gives the equation of geodesics: χ̈ q +

g qk (∂ i g kj + ∂ j g ik − ∂ k g ij )χ̇ i χ̇ j = 0 , 2

q = 1, . . . , m .

(2.29)

Here, [g ij ] = [g ij ]−1 . Remark 2.8. In order to derive the equation of geodesics one calculates the derivatives of Lagrangian L(χ i , χ̇ i ) = √g ij (χ 1 , . . . , χ m )χ̇ i χ̇ j on two variables χ i and χ̇ i , i.e., ∂ k g ij χ̇ i χ̇ j ∂L , = 2L ∂χ k g ik χ̇ i ∂L = , L ∂ χ̇ k

∂ j g ik χ̇ i χ̇ j + g ik χ̈ i 1 ⋅ d ∂L ( k ) = ( ) g ik χ̇ i + . dt ∂ χ̇ L L

The derivation with respect to an arbitrary parameter t will lead to cumbersome expressions. In order to simplify the calculations, suppose without loss of generality that the geodesic curve can be parameterized by a natural parameter, such that ‖χ 󸀠 (t)‖g = 1. Then, for all χ i , the Lagrangian L = 1 (but not its derivative) and the Euler–Lagrange equations can be written as 1 ∂ k g ij χ̇ i χ̇ j − ∂ j g ik χ̇ i χ̇ j − g ik χ̈ i = 0 , 2

k = 1, . . . , m .

Since ∂ j g ik χ̇ i χ̇ j = (∂ j g ik + ∂ i g kj )χ̇ i χ̇ j /2, the above equation transforms into the following one: 1 (∂ k g ij − ∂ j g ik − ∂ i g kj )χ̇ i χ̇ j − g ik χ̈ i = 0 , k = 1, . . . , m . 2 Finally, multiplying both sides of this equation by g qk and summing the left-hand side over k, we obtain (2.29).

2.4.4 Parallel Transport Connection. An affine connection ∇ on P is a mapping (see Chapter 9) ∇ : Vec(P) × Vec(P) → Vec(P) ,

∇ : (u, v) 󳨃→ ∇u v .

2.4 Riemannian Space |

39

Its coefficients relative to a smooth chart on P are denoted by γ i jk . One has the following coordinate representation for ∇u v: ∇u v = u c {∂ c v b + v a γ b ca } ∂ b . A smooth curve χ : 𝕀 → P is said to be autoparallel if ∇χ󸀠 χ 󸀠 = 0. In a smooth chart, this equation reduces to the system q χ̈ q + γ ij χ̇ i χ̇ j = 0 ,

q = 1, . . . , m .

Require from the connection ∇ that autoparallel curves are exactly geodesics. This implies that in a coordinate frame (∂ i )m i=1 connection coefficients must have the following representations: g im (∂ j g mk + ∂ k g mj − ∂ m g jk ) , γ i jk = 2 where g ij = g(∂ i , ∂ j ), [g ij ] = [g ij ]−1 . Note that such an expression literally coincides with the expression (2.21) for Christoffel symbols. The connection ∇ satisfies the conditions: (i) ∀i, j, k = 1, . . . , m : γ i jk = γ i kj (torsion free); (ii) ∇g = 0, or, in components, ∂ i g jk − Γ mij g mk − Γ mik g mj = 0 (metric compatible). The connection ∇ is called the Levi–Civita connection. Parallel transport. A vector field u ∈ Vec(P) is said to be parallel transported along a smooth curve χ : 𝕀 → P, if ∇χ󸀠 u = 0. In a smooth chart on P, u̇ b + u c χ̇ a γ b ca = 0 ,

b = 1, . . . , m .

Let p = χ(t0 ). Then, one has the following initial condition: u p = u 0 . Thus, locally, there exists a unique solution for the Cauchy problem u̇ b + u c χ̇ a γ b ca = 0 , u bp

=

u 0b

b = 1, . . . , m , ,

b = 1, . . . , m .

(2.30)

One can define the parallel transport map P tt 0 : T γ(t 0) P → T γ(t)P as follows: with each element u 0 ∈ T γ(t 0) P it assigns the value u(t) of the solution u for the Cauchy problem (2.30); P tt 0 : u 0 󳨃→ u(t). Suppose that p ∈ P and χ is a smooth curve on P, such that χ(0) = p. Using the parallel transport map one can express the value of covariant derivative in terms of directional derivative: ∇u v|p := lim

h→0

󵄨󵄨 P0h v χ(h) − v χ(0) d 0 󵄨 = P t v χ(t) 󵄨󵄨󵄨 . 󵄨󵄨t=0 h dt

Here, P0t v χ(t) belongs to T χ(0) P, and it means that the difference P0t v χ(t) − v χ(0) is well defined and belongs to T χ(0) P.

40 | 2 Geometry of Physical Space

Curvature. The Riemann curvature tensor R : Vec(P) × Vec(P) × Vec(P) → Vec(P) , is defined as follows: R(u, v, w) = ∇u ∇v w − ∇v ∇u w − ∇[u,v] w . Due to the definition, the Riemann curvature tensor can be represented by the dyadic decomposition R = R t ijk ∂ t ⊗ dx i ⊗ dx j ⊗ dx k , where R t ijk = ∂ i Γ tjk − ∂ j Γ tik + Γ ljk Γ til − Γ lik Γ tjl . One can introduce the purely covariant version of R. It is represented by the dyadic decomposition R♭ = R tijk dx t ⊗ dx i ⊗ dx j ⊗ dx k , where R tijk = g tl R l ijk . The Riemann curvature tensor defines the Ricci curvature tensor Ric = R ij dx i ⊗ dx j [15], where R ij = R l ilj . In turn, the Ricci tensor defines the Ricci scalar (scalar curvature) S = g ij R ij . The geometrical meaning of the Riemann curvature tensor is the following [13, 49]. The difference between the final and the original directions of a vector that was parallel transported around an infinitesimal parallelogram is of second order and depends on the value of R t ijk at the starting point and on the curve.

2.4.5 Integration There exists the Riemannian volume form dVg on P, which belongs to the orientation defined by the volume form μ. In a smooth chart, dVg = √g dx1 ∧ ⋅ ⋅ ⋅ ∧ dx m . If S ⊂ P is some smooth compact oriented submanifold of P, then its volume, Volg (S), is calculated via Volg (S) = ∫ dV ιS∗ g , S ∗ ιS g is the pullback of g to the space S, and dV ιS∗ g

where is the corresponding volume form. Here, ιS : S → P is the inclusion map, that is, the restriction of the identity map IdP to S.

2.5 Newtonian Space-Time Formalized ways to describe the notions of space and time have attracted the interest of innovative individuals since ancient times. A retrospective vision of these concepts

2.5 Newtonian Space-Time | 41

is well expressed in [50]. One tends to assume that, from the standpoint of conventional continuum mechanics, the space is usually regarded as “absolute”, i.e., independent of the observer, 3D Euclidean space, while the time is represented by real numbers that are in accordance with a time scale depending on the chosen “absolute” clock. Nevertheless, already from the beginning of the twentieth century, i.e., almost simultaneously with the formulation of the foundations on special and general relativity, works on the four-dimensional formulation of continuum mechanics equations began to appear [35, 51]. In this respect, it is worth recalling that one of the first recovered exact solutions of the Einstein equations describes the flow of perfect relativistic fluid that represents our Universe at large scale. Moreover, in the middle of the twentieth century, it was established that 4D formalism allows concise formulation of governing equations for elastic continuum with distributed defects (this issue will be discussed in more detail below). In this regard, 4D representations are no longer unaccustomed in the framework of continuum mechanics. In the following, we briefly touch upon a few of fundamentals of both Newtonian mechanics and general relativity. This, in our opinion, allows us to be more concise in the first place, and secondly, points to the close affinity of Newtonian and Einsteinian mechanics. We begin with brief consideration of the space-time structure, accepted in conventional continuum mechanics. In common life, we recognize ourselves and surrounding objects as occupying places. Places are represented by points in a three-dimensional space. As was mentioned above, important notions of classical understanding of space are translation and congruence. These notions lead to Euclidean point space. The geometric structure of the latter allows one to define points and vectors (translations), to introduce position and displacement vector fields, to introduce a global Cartesian frame, and to identify vectors and covectors by virtue of the scalar product. Changes that we observe in ourselves and in the surrounding world are considered as occurring at specific instants. The latter are represented as points in onedimensional space, independent of the space of places. Introducing the affine-Euclidean structure and orientation on such space allows one to fix the coordinate frame and to appreciate future and past. Thus, the coordinate of an instant is time, and difference between coordinates is the time lapse.

2.5.1 Newtonian Space-Time Manifold Place and instance are combined to define an event. The collection of all events form space-time. Now, we switch to precise language. An event is considered to be a primitive entity, like a point in Euclidean geometry, and we give the following definition.

42 | 2 Geometry of Physical Space

A Newtonian space-time is a 5-tuple²⁷ (M, S, D, ∇, t) , where M is a set whose elements are referred to as events, S is a topology on M, D is a smooth structure on M, ∇ is a connection on M, and t : M → ℝ is an absolute time smooth function. The following axioms are stated: N1 ) The triple (M, S, D) is a smooth 4-dimensional manifold. N2 ) ∀p ∈ M : (dt)p ≠ 0 (time never stops). N3 ) ∀p ∈ M : (∇dt)p = 0 (time flows uniformly). N4 ) ∇ is torsion free. Hereinafter the symbol d in N2 ) and N3 ) denote the exterior differentiation operation²⁸. We assume that t(M) = ℝ. For any τ ∈ ℝ denote Sτ = {p ∈ M | t(p) = τ} . The axiom N2 ) states that Sτ1 and Sτ2 with distinct τ1 , τ2 ∈ ℝ do not intersect. Thus, M is represented as the disjoint union M = ⨆ Sτ . τ∈ℝ

Each Sτ is called the absolute space at time τ. Remark 2.9. The absolute space may have a Euclidean affine structure (conventional 3D mechanics of a continuum) or may have a Riemannian structure (2D mechanics of material surfaces). Suppose that p ∈ M. By virtue of N2 ) one can make the following classification of tangent vectors. A vector u ∈ T p M is called ∙ future directed, if dt(u) > 0; ∙ spatial, if dt(u) = 0; ∙ past directed, if dt(u) < 0.

2.5.2 Newton’s Laws Smooth curves on M are called world lines of particles. Since one knows ∇, the autoparallel world line can be determined. This curve plays the role of a “straight line”, on which a particle moves uniformly when no forces act. The Newton I is formalized as

27 The present and subsequent paragraphs are based on Frederic P. Schuller’s Lecture 9 at the International Winter School on Gravity and Light (2015) https://www.youtube.com/watch?v=IBlCu1zgD4Y. 28 See Chapter 13, Section 13.3.9.

2.5 Newtonian Space-Time | 43

follows: the world line of a particle with the influence of no force is an autoparallel curve with future directed velocity at every point. If χ is such world line, then, by definition: i) ∇χ󸀠 χ 󸀠 = 0; ii) dt(χ 󸀠 ) > 0. Suppose that χ is a world line. The Newton II law states that the deviation of ∇χ󸀠 χ 󸀠 = 0 from zero is caused by forces that act on the particle. Formally, 1 f, m where f ∈ Vec(M) is a force that acts on the particle. It is assumed to be a spatial vector field, i.e., dt(f) = 0, since the force is supposed to accelerate in spatial directions. The coefficient m is a positive constant, a mass of the particle. Thus, ∇χ󸀠 χ 󸀠 is the acceleration of the particle. Assume that one can choose an atlas A from the smooth structure D that has the following form. For any chart (U, φ), a coordinate mapping φ : x 󳨃→ (x0 , x1 , x2 , x3 ) is such that φ0 : x 󳨃→ x0 coincides with the absolute time function t: φ0 = t|U . The postulate N3 ) takes the form ∇χ󸀠 χ 󸀠 =

∇∂ i dx0 = 0 ,

i = 0, 1, 2, 3 ,

which, since (∇∂ i dx0 )j = −Γ 0ji , means that Γ 0ji = 0, for all i, j = 0, 1, 2, 3. Such an atlas is called stratified. Consider the Newton II law in the atlas A: (χ 0 )󸀠󸀠 + Γ 0ij (χ i )󸀠 (χ j )󸀠 = 0 , (χ q )󸀠󸀠 + Γ

q

ij (χ

i 󸀠

) (χ j )󸀠 =

1 q f , m

q = 1, 2, 3 .

The right-hand side of the first equation is zero, since F is spatial. Since Γ 0ij = 0, one has (χ 0 )󸀠󸀠 = 0. Thus, there are constants a, b ∈ ℝ, such that χ 0 (λ) = aλ + b. The next three equations can be represented as follows: 1 q f , m where q, γ, δ = 1, 2, 3. Here, the postulate N4 ) was used. Now, parameterize the world line by the absolute time. One has d/dλ = ad/dτ. Changing primes by dots one can obtain 1 q q q q χ̈ q + Γ γδ χ̇ γ χ̇ δ + Γ 00 χ̇ 0 χ̇ 0 + 2Γ γ0 χ̇ γ χ̇ 0 = f , q, γ, δ = 1, 2, 3 . ma2 The left-hand side of this equation is the q-th component of the acceleration vector. None of the terms of this side transform as a vector, but they have the following meaning: q ∙ Γ γδ χ̇ γ χ̇ δ is a coordinate correction term; q ∙ Γ 00 χ̇ 0 χ̇ 0 is gravity or centrifugal term; q ∙ 2Γ γ0 χ̇ γ χ̇ 0 is Coriolis term. (χ q )󸀠󸀠 + Γ

q

γδ (χ

γ 󸀠

) (χ δ )󸀠 + Γ

q 0 󸀠 0 󸀠 00 (χ ) (χ )

+ 2Γ

q

γ0 (χ

γ 󸀠

) (χ 0 )󸀠 =

44 | 2 Geometry of Physical Space

2.5.3 Rigid Frame A rigid frame [52] is a bijective mapping φ : M → E×𝕋, whereE is a three-dimensional Euclidean affine space and 𝕋 ⊂ ℝ is an open interval, referred to as a chronometric manifold. We presume that φ satisfies two conditions²⁹: (i) for any t ∈ 𝕋 there exists s ∈ 𝕋, such that φ(St ) = E × {s}; (ii) for any s ∈ 𝕋 there exists t ∈ 𝕋, such that φ−1 (E × {s}) = St . Remark 2.10. We claimed that the chronometric manifold 𝕋, whose points correspond to instants of time (time points), is assumed to be diffeomorphic to ℝ. To justify the correspondence of this hypothesis to our relationship to the physical world, we follow the reasonings of H. Weyl [35]. The real world appears to us as a sequence of intentional objects of the human mind. In this regard, time can be interpreted as the primary form (Urform) of the flow of observations. Because of the nature of human consciousness, this flow is continuous and directed from the “past” to the “future”. Furthermore, in the observable world, there are processes that occur (to our perception) cyclically. These cyclic processes, which might be termed as a clock, allow one to calibrate the flow of observations, i.e., perform its arithmetization. Similar considerations (reflecting the philosophical position of intuitionism) lead to a hypothesis 𝕋 ≅ ℝ. More strictly, 𝕋 can be identified with an one-dimensional vector space over the field ℝ, the only element of whose basis is a vector characterizing the unit of time (which is reflected, in particular, in its “physical dimension”).

2.5.4 Change of Frame Let

φ: M → E × 𝕋 ,

φ : e 󳨃→ (x, t) ,

∗

φ∗ : e 󳨃→ (x∗ , t∗ ) ,

φ : M →E×𝕋, be two rigid frames. Then the bijection

φ12 = φ∗ ∘ φ−1 : E × 𝕋 → E × 𝕋 ,

(x, t) 󳨃→ (x∗ , t∗ )

is a change of frame [52]. In component functions, it is represented as (x∗ , t∗ ) = φ12 (x, t) = (α(x, t), β(t)) , where α : E × 𝕋 → E, and β : 𝕋 → 𝕋. Remark 2.11. The value of the mapping φ12 at any element (x, t) ∈ E × 𝕋 has the form ̃ t)) , φ12 (x, t) = (α(x, t), β(x,

29 In each rigid frame simultaneous events stay simultaneous.

2.5 Newtonian Space-Time |

45

where α : E×𝕋 → E and β̃ : E ×𝕋 → 𝕋 are component functions. Factually, β̃ does not depend on x. Fix t ∈ 𝕋 and take arbitrary x, x󸀠 ∈ E. Since the points (x, t), (x󸀠 , t) are simultaneous, according to (ii) there exists τ ∈ 𝕋 for which φ−1 (x, t), φ−1 (x󸀠 , t) ∈ Sτ . According to (i), for φ∗ there exists s t ∈ 𝕋, such that φ12 (x, t), φ12 (x󸀠 , t) ∈ E × {s t }. This implies ̃ t) = β(x ̃ 󸀠 , t) = β(t) , β(x, where β : 𝕋 → 𝕋, β : t 󳨃→ s t . Consider a special type of change of frame. This type is defined by Galilean transformations and based on the assumption that distances between simultaneous events and time lapses between events do not depend on a rigid frame. Remark 2.12. The notions of the rigid frame and the Galilean transformation group, which generates a change of frame, are necessary to postulate the principle of material frame-indifference. The latter principle gives certain physical restrictions on possible constitutive relations. Let e, f ∈ M be two simultaneous events. A distance d(e, f; φ) between e and f relative 󳨀 → 󳨀 → to a rigid frame φ : M → E × 𝕋 is a Euclidean distance √ xy ⋅ xy, where (x, t) = φ(e), (y, t) = φ(f). That is, 󳨀 → 󳨀 → d(e, f; φ) = √xy ⋅ xy . Let e, f ∈ M be two events. A time lapse T(e, f; φ) between e and f relative to a rigid frame φ : M → E × 𝕋 is the difference t2 − t1 , where (x, t1 ) = φ(e), (y, t2 ) = φ(f). That is, T(e, f; φ) = t2 − t1 . We state the following postulates: (P1 ) Let e, f ∈ M be two simultaneous events. For any rigid frames φ, φ∗ , the following equality holds: d(e, f; φ) = d(e, f; φ∗ ) . (P2 ) Let e, f ∈ M be two events. For any rigid frames φ, φ∗ , the following equality holds: T(e, f; φ) = T(e, f; φ∗ ) . From (P1 ) and (P2 ) it follows that for any change of frame, (x, t) 󳨃→ φ12 (x, t) = (x∗ , t∗ ) ,

46 | 2 Geometry of Physical Space there exists a mapping a : 𝕋 → E, an origin b ∈ E, a field Q : 𝕋 → O(3) of orthogonal tensors³⁰, and a number c ∈ 𝕋, such that [52] x∗ = a(t) + Q(t)(x − b) , ∗

t =t−c.

(2.31) (2.32)

2.6 Relativistic Space-Time We have come to the most complicated model of physical space considered in this book, but at the same time, to the most bewitching of them. This is relativistic spacetime. In order to become familiar with it, one has to extend the definition of Riemannian space given above somewhat.³¹

2.6.1 Lorentzian Manifolds Suppose that M is a smooth m-dimensional manifold. A pseudo-Riemannian metric³² is a section g ∈ Sec(T ∗ M ⊗ T ∗ M), such that for every point p ∈ M the following axioms hold: L1 ) ∀u, v ∈ T p M : gp (u, v) = gp (v, u); L2 ) ∀u ∈ T p M : (∀v ∈ T p M : gp (u, v) = 0) ⇔ (u = 0). In a smooth chart (U, φ), the pseudo-Riemannian metric is represented as g = g ij dx i ⊗ dx j . At each point p ∈ M, the matrix [g ij (p)] has m real non-zero eigenvalues. The amounts m + , m− of positive and negative eigenvalues are independent of the choice of p; their sum is equal to m+ + m− = m. The pair (m+ , m− ) is thus point independent and is called the signature of the metric g. In the framework of such a definition, a Riemannian metric is a pseudo-Riemannian metric with signature (m, 0). Suppose that M is a smooth four-dimensional manifold. A pseudo-Riemannian metric g with signature (1, 3) is called a Lorentzian metric. The signature for the Lorentzian metric is often denoted by (+, −, −, −). Let p ∈ M be a point. One can

30 Recall that orthogonal tensor is such a linear map that preserves inner product; see Chapter 12, Section 12.5.4. 31 The present and subsequent paragraphs are based on Frederic P. Schuller’s Lectures 13–15 in the International Winter School on Gravity and Light (2015) https://www.youtube.com/watch?v= iFAxSEoj6Go. 32 If one compares the definition of pseudo-Riemannian metric with the definition of Riemannian metric (see Chapter 13, Section 13.3.10) then one observes that the properties L1 ) and g1 ) are identical, but the properties g2 ) and g3 ) have changed to L2 ). That is, a pseudo-Riemannian metric is a field of bilinear symmetric and non-degenerate forms. These forms are not positive definite, in general.

2.6 Relativistic Space-Time | 47

choose a basis (e i )3i=0 in T p M with the property³³: gp (e i , e j ) = η ij ,

where

[η ij ] = diag {1, −1, −1, −1} .

For any vector u ∈ T p M, u = u i e i , one has gp (u, u) = (u 0 )2 − (u 1 )2 − (u 2 )2 − (u 3 )2 . Now, consider the vector space ℝ4 of all 4-tuples (u 0 , u 1 , u 2 , u 3 ). The set C T p M = {u ∈ T p M | gp (u, u) = 0} is represented in ℝ4 by the set Cℝ4 = {(u 0 , u 1 , u 2 , u 3 ) ∈ ℝ4 | (u 0 )2 − (u 1 )2 − (u 2 )2 − (u 3 )2 = 0} , which is a circular double cone with the apex (0, 0, 0, 0).

2.6.2 Time Orientation Suppose that M is a smooth four-dimensional manifold with the Lorentzian metric g. The field g establishes the distribution {C T p M }p∈M of double cones in tangent spaces. Recall that in the case of Newtonian space-time, one has the absolute time function t. The condition dt(u) = 0 generates a hyperplane in tangent space, and future directed vectors are situated on one side of it. Thus, the absolute time function allows us to specify a distribution of half-spaces of tangent spaces. In relativistic physics, no absolute time is presupposed. According to one of relativistic postulates, velocities of mass particles must be situated inside the cones in the future directed positions. In other 1/2 words, one needs a rule that allows one to specify a distribution {C T p M }p∈M of circular cones from the given distribution {C T p M }p∈M of double cones. This means that for every point p ∈ M, one needs to choose exactly one of two cones that constitute C T p M . This choice can be made using time orientation. A time orientation on M is a vector field T ∈ Vec(M) with the properties [53]: T 1 ) T ≠ 0 T2 ) g(T, T) > 0. By definition, for each point p ∈ M the value T(p) belongs to one of two cones that con1/2 stitute C T p M . Thus, it is exactly the desired rule of specifying a distribution {C T p M }p∈M of circular cones. The role of the absolute time function is now played by the pair (g, T) of the Lorentzian metric and time orientation.

33 In the case of Lorentzian manifolds, we consider values of indices from the set {0, 1, 2, 3}.

48 | 2 Geometry of Physical Space

2.6.3 Definition of Relativistic Space-Time A relativistic space-time is a 7-tuple [31, 33] (M, S, D, g, ∇, T, μ) , where M is a set whose elements are referred to as events, S is a topology on M, D is a smooth structure on M, g ∈ Sec(T ∗ M ⊗ T ∗ M) is a section, ∇ is a connection on M, T ∈ Vec(M) is a vector field on M, and μ is a volume form on M. The following axioms are stated: R1 ) The triple (M, S, D) is a smooth four-dimensional manifold. R2 ) g is a Lorentzian metric. R3 ) ∇ is the Levi–Civita connection. R4 ) T is a time orientation. Smooth curves on M are called world lines of particles. In contrast to Newtonian space-time, the relativistic space-time admits massive and massless particles. They are subjected the following two postulates: P1 ) The world line χ : 𝕀 → M of a massive particle satisfies: (i) ∀t ∈ 𝕀 : gχ(t)(χ 󸀠 (t), χ 󸀠 (t)) > 0 (i.e., the velocity lies inside the double cone C T χ(t) M ); (ii) ∀t ∈ 𝕀 : gχ(t) (T t , χ󸀠 (t)) > 0 (i.e., the velocity lies exactly in that cone which is specified by T). P2 ) The world line χ : 𝕀 → M of a massless particle satisfies: (i) ∀t ∈ 𝕀 : gχ(t) (χ 󸀠 (t), χ 󸀠 (t)) = 0 (i.e., the velocity lies in the boundary of the double cone C T χ(t) M ); (ii) ∀t ∈ 𝕀 : gχ(t) (T t , χ󸀠 (t)) > 0 (i.e., the velocity belongs to the boundary of such cone which is specified by T).

2.6.4 Observers Both a time and the velocity of a particle are observer-dependent quantities. This notion is formalized as follows. An observer is a world line γ : 𝕁 → M of a massive particle together with a choice of the basis (e 0 (t), e1 (t), e2 (t), e3 (t)) on each T γ(t) M, t ∈ 𝕀. This collection of bases needs to satisfy the following conditions: (i) ∀t ∈ 𝕀 : e0 (t) = γ󸀠 (t); (ii) ∀t ∈ 𝕀 : gγ(t)(e i (t), e j (t)) = η ij , i, j = 0, 1, 2, 3.

2.6 Relativistic Space-Time |

49

Any observer is subjected to the following postulates: P1 ) A clock carried by a specific observer (γ, e) will measure a time t2

τ := ∫ √gγ(t)(γ󸀠 (t), γ󸀠 (t)) dt t1

between the two “events”: ∙ γ(t1 ) – “start the clock”; ∙ γ(t2 ) – “stop the clock”. The number τ is called the proper time. P2 ) Let (γ, e) be an observer and χ : 𝕀 → M be a massive particle world line, which is parameterized such that ∀t ∈ 𝕀 : gχ(t)(χ 󸀠 (t), χ 󸀠 (t)) = 1. Suppose that the observer and the particle meet somewhere in space-time. That is, χ(t1 ) = γ(t2 ), for some t1 ∈ 𝕀, t2 ∈ 𝕁. The observer measures the 3-velocity (spatial velocity) of this particle as v := e αt2 (χ 󸀠 (t1 )) e α (t2 ) , α = 1, 2, 3 , where (e it 2 )3i=0 is the unique dual basis of (e i (t2 ))3i=0 .

2.6.5 Lorentz Transformations Let (γ, e) and (̃γ , ̃e) be observers with ̃γ(̃t) = p = γ(t). Reparametrizing the curves, if necessary, one may assume that ̃t = t = 0. Thus, (e i (0))3i=0 and (̃e i (0))3i=0 are bases of the similar tangent space T p M. There exists a matrix Ω ∈ GL(4; ℝ), such that ̃e i (0) = Ω j i e j (0) . Since gp (̃e i (0), ẽ j (0)) = η ij , gp (e k (0), e l (0)) = η kl and using the relation between “twiddled” and “non-twiddled” vectors, one obtains the Lorentz transformation Ω k i Ω l j η kl = η ij . Thus, the Lorentz transformation relates the frames of two observers at the same point. One has Ω ∈ O(1, 3).

2.6.6 Matter There are two types of matter: point matter and field matter. Assume that g and T are given. Point matter. Postulates P1 ) and P2 ) about massive and massless particles give constraints on the possible particle world lines. To obtain world lines one needs to use law of motion. Consider actions of particles in the case of no acting external fields.

50 | 2 Geometry of Physical Space

The action of a massive particle with world line χ has the form Amassive [χ] = m ∫ √gχ(t) (χ 󸀠 (t), χ 󸀠 (t)) dt . Here, m > 0 is a constant, a mass of the particle. Dynamical laws of motion (Euler – Lagrange equations) are obtained from the condition δAmassive [χ] = 0. The possible world lines χ need to satisfy the restriction gχ(t)(T t , χ󸀠 (t)) > 0, for all t ∈ 𝕀. The action of a massless particle with world line χ has the form Amassless [χ, λ] = ∫ λ(t)√gχ(t) (χ 󸀠 (t), χ 󸀠 (t)) dt . Here, λ(t) is a Lagrange multiplier. The partial variation of Amassless [χ, λ] with respect to λ gives the condition gχ(t)(χ 󸀠 (t), χ 󸀠 (t)) = 0, for all t. Dynamical laws of motion (Euler – Lagrange equations) are obtained from the partial variation of Amassless [χ, λ] with respect to χ. The reason for describing equations of motions by actions is that composite systems have an action that is the sum of the actions of the parts of this system possibly including interaction terms. Consider an example. Suppose that a massive particle interacts with the electromagnetic field that is generated by a covector field A ∈ CVec(M) (electromagnetic potential). Let q be a charge of this field. Then, the action has the form A[χ; A] = ∫ (m√gχ(t) (χ 󸀠 (t), χ 󸀠 (t)) + qA(χ 󸀠 (t)) dt . Here, the semicolon means that A is fixed. The Euler–Lagrange equation has the form m(∇χ󸀠 χ 󸀠 )i = −qF ji χ̇ j , ∂A i where F ij = ∂A − ∂x ij are components of the Faraday tensor. The expression −qF ji χ̇ j ∂x j represents the Lorentz force on a charge particle in the electromagnetic field. Note that the action A[χ; A] has the form

A[χ; A] = Amassive [χ] + Ainteract [χ; A] , where Ainteract [χ; A] is the interaction term. Field matter. A classical field matter is any tensor field on M whose equations of motions are derived from an action. Consider an example. The action for electromagnetic field, in assumption that M is a trivial manifold, has the form (in the atlas with one chart) 1 AMaxwell [A] = ∫ √−gF ab F cd g ac g bd d4 x , 4 M

where g = det[g ij ] < 0.

2.6 Relativistic Space-Time |

51

Remark 2.13. Note that the Lagrangian density of AMaxwell depends on the first partial derivatives of the electromagnetic potential A. A similar situation holds for simple materials in continuum mechanics. The response functional for the simple material depends on the deformation gradient which, in a basis, is represented by first-order partial derivatives of actual coordinates. The energy-momentum tensor of matter fields. Suppose that the metric g is unknown, and one intends to obtain it. Thus, one needs to write down the action Agrav[g] for the metric tensor. This action will be added to any action Amatter [A, . . . ] in order to describe the total system. In particular, A[A, g] = Agrav [g] + AMaxwell [A, g] . The partial variation of A[A, g] with respect to δA gives the Maxwell equations. Only AMaxwell gives contributions for it. In the case of the partial variation of A[A, g] with respect to δg, both Agrav and AMaxwell give contributions. Thus, the Euler–Lagrange equations have the form 1 G ij 16πGNewt ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ contribution from Agrav

+

1 (− T ij ) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2

=0,

contribution from AMaxwell

where GNewt is Newton’s gravitation constant. In another form, this equation is represented as G ij = 8πGNewt T ij . This is the Einstein equation. The quantities T ij form the energy-momentum tensor. In general, if Amatter [Φ, g] is some matter action with the Lagrangian density Lmatter , then the energy-momentum tensor is T ij := −

∂Lmatter 2 ∂Lmatter − ∂s + ⋅⋅⋅) . ( ∂g ij ∂∂ s g ij √−g

2.6.7 The Einstein Equation In Newtonian space-time, Poisson’s law ∆φ = 4πρGNewt can be formulated in terms of the Ricci curvature as R00 = 4πρGNewt . Based in this equation, Einstein suggested that the relativistic field equations for the metric g of relativistic space-time should be as follows: R ij = 8πGNewt T ij . However, this equation is wrong because T ij are components of the energy-momentum tensor that is derived from an action. The divergence of the right-hand side of the

52 | 2 Geometry of Physical Space

equation is zero. The divergence of the left-hand side of the equation is not zero, in general, which leads to a contradiction. These equations cannot be upheld. In order to find the appropriate left-hand side of the equation Hilbert suggested considering the action AHilbert [g] = ∫ √−gR ij g ij d4 x . M

The stationary points of the functional AHilbert can be found as the solutions of the corresponding Euler–Lagrange equations. In order to obtain them, derive the variation of AHilbert : δAHilbert [g] = ∫ {δ√−gR ab g ab + √−gR ab δg ab + √−gg ab δR ab } d4 x .

(2.33)

M

Using formula for derivative of the determinant (see the derivation of the Voss–Weyl formula), we obtain δ√−g =

1 1 (−g)g ab δg ab = √−gg ab δg ab . 2√−g 2

Taking into account the identity g ab g bc = δ ac , which leads to the following equation: δ(g ab )g bc + g ab δg bc = 0 , we arrive at δg ab = −g am g bn δg mn . In order to simplify further derivations suppose that the coordinates, defined by the chosen chart, are normal with respect to point p. Then, all Γ i jk (but not their derivatives) vanish at p. Thus, the variation of the Ricci tensor can be calculated as follows: δR ab = δ∂ b Γ mam − δ∂ m Γ mab = ∂ b δΓ mam − ∂ m δΓ mab = ∇b δΓ mam − ∇m δΓ mab , because δ and ∂ commute, all terms including Γ’s are reduced to zero, and partial derivatives (at point p) coincide with corresponding covariant derivatives. Keeping in mind that M is a pseudo-Riemannian manifold, and, consequently, ∇c g ab = 0, we obtain √−gg ab δR ab = √−gg ab ∇b δΓ mam − √−gg ab ∇m δΓ mab = √−g∇b (g ab δΓ mam ) − √−g∇m (g ab δΓ mab ) . Finally, using the Voss–Weyl formula (2.22), we get √−gg ab δR ab = ∂ b (√−gδA b ) − ∂ m (√−gδB m ) = ∂ b (√−g(δA b − δB b )) , where δA b = g ab δΓ mam , δB b = g am δΓ bam .

2.6 Relativistic Space-Time | 53

Collecting all items in (2.33) one arrives at the following expression: 1 δAHilbert [g] = ∫ { √−gg mn δg mn R ab g ab − √−gg am g bn δg mn R ab 2 M

+ ∂ b (√−g(δA b − δB b ))} d4 x 1 = ∫ {√−gδg mn ( g mn S − R mn )} d4 x + ∫ ∂ b (√−g(δA b − δB b )) d4 x . 2 M

M

Here, S is the scalar curvature. The last item is an integral of divergence and, following standard argumentation (see [32]), we put it equal to zero. Thus, in view of the functional independence of g mn , 1 (δAHilbert [g] = 0) ⇔ ( g mn S − R mn = 0) . 2

(2.34)

We use the following notation for the Einstein tensor: G ij = R ij −

1 ij g S, 2

G ij = g ia g jb G ab .

Then, the equations (2.34) can be rewritten as ∇i G ij = 0 .

(2.35)

Note that from a purely geometrical point of view, the Einstein tensor satisfies the Bianchi identity, which formally coincides with (2.35). If the space is filled with interacting matter, then the action AHilbert is more complex. For example, if we take into account electromagnetic interaction, then AHilbert depends on the metric and the electromagnetic potential, which leads to the so-called Einstein–Maxwell theory. Thus, in the case of interacting matter, the equations are the following: 1 R ij − Sg ij = 8πGNewt T ij . (2.36) 2 These equations were obtained by Hilbert. Using physical arguments, Einstein came to similar equations [54]. The equations (2.36) are called the Einstein equations, and the action AHilbert is called the Einstein–Hilbert action. Einstein equations are partial differential equations. Some particular solutions are given below. The Schwarzschild solution. Let M be a flat (Minkowski) space equipped with the pseudo-Cartesian chart (x0 , x1 , x2 , x3 ) and let B be an open 3D ball of radius R with center at the origin of the pseudo-Cartesian frame (o, (i k )3k=0 ), i.e., M ⊃ B := {o + x1 i1 + x2 i2 + x3 i3 | √(x1 )2 + (x2 )2 + (x3 )2 < R} . Let U = M \ B be an open subset of M that represents the exterior of B. Following K. Schwarzschild, find the curvature of M caused by the filling of the region B

54 | 2 Geometry of Physical Space

with non-interacting matter (dust) of total mass M. To do this, we use a semi-inverse method. Keeping in mind spherical symmetry with respect to spatial coordinates choose the chart (U, h) on M that relates with the pseudo-Cartesian chart by the following transition functions: x0 = t , x1 = r cos φ sin θ ,

(2.37)

x2 = r sin φ sin θ , x3 = r cos θ .

The components of the metric tensor induced by (2.37) and its inverse have the form ∘

g 00 = 1 , ∘

g 00 = 1 ,

∘

g 11 = −1 , ∘

g 11 = −1 ,

∘

g 22 = −r2 , ∘ 1 g 22 = − 2 , r

∘

g 33 = −r2 sin2 θ , ∘ 1 g 33 = − . 2 r sin2 θ

∘

The square of the interval, d s 2 , can be written as (hereinafter we consider speed of light c = 1) ∘

g = dt ⊗ dt − dr ⊗ dr − r2 dθ ⊗ dθ − r2 sin2 θdφ ⊗ dφ .

(2.38)

Now, assume that the region B is filled with massive non-interacting matter that brings flat M to some curved space-time. In order to determine the metric of curved M, construct the generalization of (2.38), g = Udt ⊗ dt − Vdr ⊗ dr − Wr2 dθ ⊗ dθ − Xr2 sin2 θdφ ⊗ dφ , where U, V, W, X are some functions of coordinates. Taking into account (i) spherical symmetry with respect to spatial coordinates (ii) the static character of problem we conclude that U, V, W, X do not depend on θ, φ; moreover, W = X = 1, and these functions do not depend on t. Thus, ds2 = U(r)dt ⊗ dt − V(r)dr ⊗ dr − r2 dθ ⊗ dθ − r2 sin2 θdφ ⊗ dφ . Consequently, g00 = U(r) , 1 , g00 = U(r)

g22 = −r2 , 1 g22 = − 2 , r

g11 = −V(r) , 1 g11 = − , V(r)

g33 = −r2 sin2 θ , 1 g33 = − . 2 r sin2 θ

(2.39) (2.40)

For the given metric components (2.39), we can calculate connection coefficients Γ ijk : Γ 001 = Γ 010 =

U󸀠 , 2U

U󸀠 , 2V r = − sin2 θ , V 1 = Γ 331 = , r

V󸀠 , 2V

Γ 100 =

Γ 111 =

r , V

Γ 133

Γ 212 = Γ 221 =

Γ 233 = − cos θ sin θ ,

Γ 313

Γ 122 = −

1 , r

Γ 323 = Γ 332 = cot θ .

(2.41)

2.6 Relativistic Space-Time |

55

Now, we can derive the components of the Ricci tensor: U 󸀠󸀠 U 󸀠 V 󸀠 (U 󸀠 )2 1 U 󸀠 + − , + 2V 4UV r V 4V 2 U 󸀠󸀠 (U 󸀠 )2 U 󸀠 V 󸀠 1 V 󸀠 = − − − , 2U 4UV r V 4U 2 1 rV 󸀠 rU 󸀠 + − = −1, 2UV V 2V 2 = R22 sin2 θ ,

R00 = − R11 R22 R33

and the scalar curvature R = g ij R ij , S=−

(U 󸀠 )2 2 U 󸀠 2 1 U󸀠 V󸀠 2 V󸀠 U 󸀠󸀠 + + 2 (1 − ) . + + − 2 2 2 UV 2UV V 2U V r UV r V r

The tensorial Einstein equation reduces to the following system of three non-linear equations: 1 V󸀠 1 1 + 2 (1 − ) = 0 , r V V r U󸀠 1 1 (2.42) − (1 − ) = 0 , + rUV r2 V r(U 󸀠 )2 U 󸀠 V 󸀠 rU 󸀠󸀠 rU 󸀠 V 󸀠 + − + + =0, − U V U 2 UV 2U 2 which admits the solution³⁴ V=

1 , 1 − A/r

U = B (1 −

A ) , r

where A and B are the constants of integration. These constants have to be chosen such that obtained metric in weak-field approximation corresponds to the Newtonian law of gravity [55]: A = 2MGNewt , B = 1 . Thus, the Schwarzschild metric g has the form g = (1 −

2MGNewt 2MGNewt −1 ) dt ⊗ dt − (1 − ) dr ⊗ dr − r2 (dθ ⊗ dθ + sin2 dφ ⊗ dφ) . r r

Note that Schwarzschild metric is (i) spherically symmetric; (ii) asymptotically flat (when r → ∞ it tends to flat space-time, Minkowski, metrics). The constant A is denoted by r s and is called the Schwarzschild radius. That is, r s = 2MGNewt .

34 The function V can be found separately from the first equation of (2.42), while the function U is determined as the solution of the second equation of (2.42), with V already found. It is easy to check that the so-obtained functions U and V satisfy the third equation (2.42).

56 | 2 Geometry of Physical Space

The Kerr solution. Assume that there is a massive body with mass M that rotates with angular momentum J. Introduce the coordinates (t, r, θ, φ), which are related with pseudo-Cartesian coordinates by the following expressions [56]: x0 = t , x1 = √r2 + a2 cos φ sin θ ,

(2.43)

x2 = √r2 + a2 sin φ sin θ , x3 = r cos θ ,

where a = J/M. In coordinates (t, r, θ, φ) (2.43), the Kerr metric is written as [57] g = (1 −

rs r ρ2 ) dt ⊗ dt − dr ⊗ dr − ρ 2 dθ ⊗ dθ 2 ∆ ρ

− (r2 + a2 +

r s ra2 r s ra sin2 θ) sin2 θdφ ⊗ dφ + 2 sin2 θ(dφ ⊗ dt + dt ⊗ dφ) , 2 ρ ρ

where r s is the Schwarzschild radius, and ρ 2 = r2 + a2 cos2 θ ,

∆ = r2 − r s r + a2 .

j

Introduce the tetrad fields h i = h i ∂ j and h i = h i j dx j with the property g(h i , h j ) = η ij , i, j = 0, 1, 2, 3. That is, η ij = g kl h i k h j l . One has the following inverse relation: g ij = η kl h k i h l j . Using this relation one can find the components of the Kerr tetrad [57]: γ00 0 [h i j ] = ( 0 0

0 γ11 sin θ cos φ γ11 sin θ sin φ γ11 cos θ

0 γ22 cos θ cos φ γ22 cos θ sin φ −γ22 sin θ

η −β sin φ ) , β cos φ 0

where γ00 = √g00 , γ ii = √−g ii , β 2 = η2 − g33 and η = g03 /γ00 . Next,

j

[h i ] = (

γ−1 00 03 −βg sin φ βg03 cos φ 0

0 γ−1 sin θ cos φ 11 γ−1 sin θ sin φ 11 γ−1 cos θ 11

0 γ−1 cos θ cos φ 22 γ−1 cos θ sin φ 22 −γ−1 sin θ 22

0 −β −1 sin φ ) . β −1 cos φ 0

In the framework of teleparallel gravity a gravitation is represented by torsion, not by curvature. The Weitzenböck connection gives an example of the affine connection,

2.7 Concluding Remarks

|

57

which gives non-zero torsion but zero curvature and non-metricity³⁵. Its connection coefficients are related with the tetrad (h i ) as Γ ijk = h l i ∂ k h l j .

(2.44)

Let us draw attention to the structure of the expression (2.44). Such a structure arises hereinafter in the definition of material connection (Chapter 9, Section 9.2.7). The role tetrads play is local deformations (implants). It also should be noted that both mentioned expressions are a particular case for the general one, which defines the calibration procedure for connection on the principal bundle (Chapter 14, Section 14.3.4). The Reissner–Nordström metric. Suppose that charged spherical massive body is given. Then, in coordinates (t, r, θ, φ), one has −1

g = (1 −

2 2 rs rq rs rq + 2 ) dt ⊗ dt − (1 − + 2) r r r r

dr ⊗ dr − r2 (dθ ⊗ dθ + sin2 dφ ⊗ dφ) , 2

G Newt . Here, q is the charge and where r s is the Schwarzschild radius and r2q = q 4πε 0 1/(4πε0 ) is Coulomb’s constant. The list of exact solutions of the Einstein equation could be continued, however, the above collection is sufficient for our purposes. Note the close affinity between them and exact universal solutions, obtained for non-linear equations of elasticity (see Chapter 3, Section 3.10).

2.7 Concluding Remarks In the rest of the book, a physical space P is a set, whose elements X, x, Y, y, . . . , are referred to as places. We suppose that such a set is endowed with Riemannian space structure (P, S, D, g, μ). It is assumed that dim P = m ≤ 3. Coefficients of the Levi–Civita connection ∇ are denoted by γ i jk . Intuitively, the physical space represents an “ideal laboratory”, where one observes places and provides measurements. The particular case of the physical space is the Euclidean affine space E introduced in 2.1.

35 The Weitzenböck connection is introduced by applying the moving frame method to tetrad (h i ). See Chapter 9, Section 9.2.6 for details.

3 Essentials of Non-Linear Elasticity Theory 3.1 Shapes and Deformation In the framework of conventional continuum mechanics, a physical space is modeled by a three-dimensional Euclidean affine space E with the translation vector space V and the inner product (⋅). Solids are modeled as subsets of E with special properties. As a rule, one requires these subsets to be open in E and to have regular boundary¹ (in the sense of O. D. Kellogg [58]). In the present chapter, such subsets are called shapes. Let us consider their properties in more detail. Shapes. Recall that, since E is Euclidean affine space, one can define the metric d : E × E → ℝ according to (2.6). Thus, (E, d) is a metric space and it makes sense to consider open subsets. The notion of the regular boundary is introduced sequentially, beginning from one-dimensional sets [58]: 1. A regular arc is a set a ⊂ E, such that for some Cartesian coordinate system it admits a representation a = {x ∈ E | x2 = f(x1 ), x3 = g(x1 ), x1 ∈ [a, b]} , 2.

3.

where a < b and f, g ∈ C∞ ([a, b]; ℝ). A regular curve is a set c ⊂ E, which consists of a finite number of regular arcs arranged in order and such that the terminal point of each arc (other than the last one) is the initial point of the following arc. The arcs have no other points in common, except that the terminal point of the last arc may be the initial point of the first one (in this case, c is a closed regular curve). A regular surface element is a non-empty set s, such that for some Cartesian coordinate system, it admits the representation s = {x ∈ E | x3 = f(x1 , x2 ), (x1 , x2 ) ∈ K} ,

4.

where K ⊂ ℝ2 is a compact connected set, such that ∂K is a closed regular curve and f ∈ C∞ (K; ℝ). The union of a finite number of regular surface elements is called a regular surface, if: (a) the intersection of any two of the elements is either empty, a single point which is a vertex for both, or a single regular arc which is an edge for both; (b) the intersection of any collection of three or more elements consists at most of vertices;

1 The requirement of regularity (in the sense of Kellogg) is due to the fact that one needs to use Stokes’ theorem for deriving the differential balance equations. https://doi.org/10.1515/9783110563214-003

60 | 3 Essentials of Non-Linear Elasticity Theory

5.

(c) any two of the elements are the first and last of a chain such that each has an edge in common with the next; (d) all elements having a vertex in common form a chain such that each has an edge, terminating in that vertex, in common with the next; the last may, or may not, have an edge in common with the first. The term “edge” here refers to one of the (finite number of) regular arcs comprising the boundary of a regular surface element, while a vertex is a point at which two edges meet. If all the edges of a regular surface each belong to two of its surface elements, the surface is a closed regular surface. A non-empty set S ⊂ E has a regular boundary, if ∂S is a closed regular surface.

All is ready for the following definition. A shape is an open non-empty set S ⊂ E, whose boundary ∂S is regular. One privileged shape SR is called a reference shape. Its points are used as “labels” in consideration of deformation. Let S be another shape; we refer to it as the actual shape. Following classical notations, we denote points of SR by the Latin majuscules X, Y, . . . and points of S by the Latin minuscules x, y, . . . . The gradient and divergence operators, which are introduced by (2.9), (2.12), and (2.13), are denoted by gradR and divR , if they are defined on SR . Deformation. A deformation is a mapping γ : SR → S ,

γ : X 󳨃→ x ,

that satisfies the following postulates [59]: (γ1 ) Postulate of impenetrability. Two particles cannot occupy the same place at the same time. (γ2 ) Any point from S is a place for some point from SR . (γ3 ) Postulate of continuity. Deformation is as many times continuously differentiable as required. The mathematical formalization of these postulates is: γ is an injective and surjective C∞ -mapping. We considered SR a non-deformed shape and S a deformed one. However, this point of view may be reversed. One can recognize S as a reference shape and SR as an actual one. Since both points of view are equal, the mapping γ−1 : S → SR should be required to be smooth. Thus, one obtains that γ is a C∞ -diffeomorphism. In the following, we use such an assumption. Finally, by a deformation we mean a C∞ -diffeomorphism γ : SR → S. The inverse diffeomorphism γ−1 : S → SR is called the inverse deformation.

3.2 Shape Coordinates and Basis

|

61

3.2 Shape Coordinates and Basis Since shapes SR and S are open subsets of E, they are open three-dimensional submanifolds of E. The scalar product (⋅) induces metric tensors G and g on SR and S. The Cartesian chart. Denote by DSR : SR → ℝ3 and DS : S → ℝ3 the restrictions of D to the shapes SR and S. Thus, {(SR , DSR )} and {(S, DS )} are atlases. The coordinate representation of γ is the following: γ d = DS ∘ γ ∘ DS−1R : D(SR ) → ℝ3 ,

(3.1)

Here, D(SR ) is an open subset of ℝ3 . The relation between γ and γ d is illustrated by the diagram E

ℝ3 ∪

∪

DS

S

D(S)

γ SR

γd DSR

D(SR )

∩

∩

E

ℝ3

In the following, a triple (X i )3i=1 denotes local coordinates induced on the shape SR by DSR , and a triple (x i )3i=1 denotes local coordinates induced on the shape S by DS . The curvilinear chart. Suppose that one introduced different curvilinear coordinate systems A1 = {(U α , σ α )}α∈I and A2 = {(V β , ψ β )}β∈J on E, which are associated with reference and actual shapes². These systems, being induced to the shapes SR and S, ̃ β )}β∈J . Here, σ ̃ α )}α∈IS and AS = {(S ∩ V β , ψ ̃ α and result in atlases ASR = {(SR ∩ U α , σ S R ̃ ψ β are restrictions of the coordinate mappings σ α and ψ β . Consider the representation of the deformation γ in these curvilinear coordinates. Let X ∈ SR . Choose α ∈ ISR , β ∈ JS , such that X ∈ U α and γ(X) ∈ V β . Denoting U αβ = U α ∩ γ−1 (V β ) one arrives at the coordinate representation 3 3 ̃β ∘ γ ∘ σ ̃ −1 γc = ψ α : ℝ ⊃ σ α (U αβ ) → ℝ .

(3.2)

2 For example, if SR is an open ball and S is an open cube, then one can introduce spherical and Cartesian coordinates on E.

62 | 3 Essentials of Non-Linear Elasticity Theory

The relation between γ and γ c is shown in the diagram E

ℝ3

∪

∪

̃β ψ

γ(U αβ )

ψ β ∘ γ(U αβ )

γ

γc ̃α σ

U αβ

σ α (U αβ )

∩

∩

E

ℝ3

Local bases and metrics. Let (Q i )3i=1 denote curvilinear coordinates induced on the ̃ α and (q i )3i=1 denote curvilinear coordinates induced on the shape S. shape SR by σ Using the relations established in Chapter 2, Section 2.2 we write Ei =

∂X k ik , ∂Q i

Ei =

∂Q i k i , ∂X k

(3.3)

for the local basis (E i )3i=1 and the dual basis (E i )3i=1 on SR , and ei =

∂x k ik , ∂q i

ei =

∂q i k i , ∂x k

(3.4)

for the local∗basis (e i )3i=1 and the dual basis (e i )3i=1 on S. The metric tensor G and its associated, G, have the decompositions G = G ij E i ⊗ E j ,

G ij = E i ⋅ E j ,

∗

G = G ij E i ⊗ E j , ∗

while the metric tensor g and its associated, g, g = g ij e i ⊗ e j ,

g ij = e i ⋅ e j ,

∗

g = g ij e i ⊗ e j . Here, [G ij ] = [G ij ]−1 and [g ij ] = [g ij ]−1 . The relation between local bases. Since the local bases (E i ) and (e i ) in (3.3) and (3.4) take values in the space V, there exists a smooth field [Ω i j ] : SR → GL(3; ℝ) of square matrices, such that j Ei = Ω i ej .

3.2 Shape Coordinates and Basis

| 63

Using the definitions of (E i ) and (e i ), one can obtain the explicit formula for [Ω i j ]. Indeed, since k ∂X k j ∂x = Ω i j , k = 1, 2, 3 , i ∂Q ∂q one obtains j

Ωi=

∂q j ∂X k . ∂x k ∂Q i

For the dual bases (E i ) and (e i ) in (3.3) and (3.4), one can similarly write E i = ℧i j e j , where [℧i j ] : SR → GL(3; ℝ) is a smooth field of matrices. From the relation ⟨E i , E j ⟩ = δ ij one obtains [℧i j ] = [Ω i j ]−1 .

The relation between representations of γ. Mappings (3.1) and (3.2) are related as follows: ̃ −1 ∘ γ c ∘ σ ̃ α ∘ DS−1 . γ d = DS ∘ ψ (3.5) β R Here,

i i ̃ −1 DSR ∘ σ α : (Q ) 󳨃→ (X ) ,

̃ −1 : (q i ) 󳨃→ (x i ) DS ∘ ψ β are transition functions. A particular case. Let ASR = {(U α , σ α )}α∈ISR be a smooth atlas on SR . The deformation γ transfers this atlas to S in such a way that points x = γ(X) have similar coordinates to points X. Formally, one can introduce the atlas AS = {(γ(U α ), ψ α )}α∈ISR on S. Here, ψ α = σ α ∘ γ−1 . The coordinate representation γ c , relative to these atlases, is just the identity map: −1 γ c = ψ α ∘ γ ∘ σ −1 α = σ α ∘ σ α = Id . Two descriptions. Lagrangian (material) and Eulerian (spatial) descriptions are traditional in conventional continuum mechanics. In the material one, the behavior of an individual point X ∈ SR is described. In this case, one uses the mappings defined on SR . In particular, the actual position of a point X ∈ SR is expressed as a value of a function of X, the deformation γ. In a spatial description, points from S are treated as independent variables, and one considers fields on S. In particular, one uses the inverse mapping γ−1 to represent points in the reference shape as values of a function defined on actual positions.

64 | 3 Essentials of Non-Linear Elasticity Theory

3.3 Deformation Gradient and Strain Measures 3.3.1 Deformation Gradient Suppose that γ : SR → S is a deformation. The deformation gradient. The deformation gradient is the mapping γ󸀠 : SR → Lin(V; V) ,

γ󸀠 : X 󳨃→ γ󸀠 (X) ,

where linear map γ󸀠 (X) is the total derivative of γ at point X ∈ SR . That is, for any h ∈ V such that X + h ∈ SR , the following expression holds (see Remark 2.3): γ(X + h) = γ(X) + γ󸀠 (X)[h] + o(‖h‖)

as

h→0.

(3.6)

Remark 3.1. In the classical works (for example, cf. [37, 45, 52]) the deformation gradient is denoted by ∇γ. We avoid such notation for the reason that the deformation gradient can be confused with the covariant derivative. Following classical notation, we will frequently use the symbols F for γ󸀠 and F X for γ󸀠 (X). In a Cartesian coordinate system (o, (i s )3s=1 ), the deformation γ has the representation γ(X) = o + γ s (X)i s for any point X ∈ SR , where γ s : SR → ℝ , are component functions. Thus, the deformation gradient has the following dyadic decomposition at point X: ∂x s 󵄨󵄨󵄨󵄨 FX = (3.7) 󵄨 is ⊗ ik . ∂X k 󵄨󵄨󵄨X In this decomposition, γ s (X + ti k ) − γ s (X) ∂x s 󵄨󵄨󵄨󵄨 lim , 󵄨󵄨 := k 󵄨 SR;X;i k ∋t→0 t ∂X 󵄨X are partial derivatives of γ d . Here, SR;X;ik = {t ∈ ℝ | t ≠ 0, X + ti k ∈ SR } is a punctured neighborhood of the point t = 0 underlying in the set SR . We usually express (3.7) as F = F sk i s ⊗ i k ,

where

F sk =

∂x s . ∂X k

Hereafter, occasionally, we will refer to the components of second-rank tensors as the elements of rectangular matrices. We will denote such matrices by placing the symbol of a tensor into square brackets. For example, [F i j ] denotes the matrix of F’s components. In order to establish correspondence between the indexed components and the

3.3 Deformation Gradient and Strain Measures

| 65

matrix elements, we adopt the convention: the first index determines a row, while second one determines a column of the matrix. For example, F 11 [F j ] = (F 21 F 31 i

F 12 F 22 F 32

F 13 F 23 ) . F 33

It should be noted that the correspondence between second-rank tensors and their matrices is not one-to-one, because the passage from tensors to matrices results in the loss of information about the type of tensor. We adopt the notation J for the determinant of F, i.e., J = det F .

(3.8)

By virtue of the injectivity of γ, ∀X ∈ SR : (J X ≠ 0) . In other words, the deformation gradient F can be considered as a field F : SR → Inv(V; V) of invertible linear mappings. Let X and Y be any points of SR . The difference Y − X is a vector, which, if Y is “close” to X, may be considered as an infinitesimal fiber in the reference state. After deformation, this fiber translates to the fiber γ(Y) − γ(X). The equality (3.6) states that the deformed fiber γ(Y) − γ(X) with accuracy up to o(‖Y − X‖) is equal to F X [Y − X]. In the following, we assume that γ satisfies the postulate: (γ4 ) Orientation preserving. For any non-coplanar fibers in SR , under action of F X an ordered triple of tangent vectors to them transforms to another ordered triple with the same orientation. In mathematical language, this requirement can be stated as ∀X ∈ SR : (J X > 0) . The deformation gradient as three 1-forms. The deformation gradient can be viewed as a triple of 1-forms [8, 60]. Suppose that γ : SR → S is a deformation. Each of the sets SR , S is an open submanifold of E. Let us identify the tangent spaces T X SR , T x S with the tangent spaces T X E, T xE (in Chapter 4, Section 4.5.4 this identification is considered in more detail), for all X ∈ SR , x ∈ S. Then, one can identify the deformation gradient F with the tangent map Tγ. Thus, F = F i j ∂ x i ⊗ dX j , and one arrives at the relation F = ∂ x i ⊗ F i , where F i = F i j dX j , i = 1, 2, 3 . (3.9) The inverse deformation gradient. The inverse deformation gradient (γ−1 )󸀠 (x) is 3 equal to F −1 X . In the Cartesian coordinate system (o, (i s )s=1 ), the following dyadic decomposition holds: ∂X m 󵄨󵄨󵄨󵄨 F −1 = i m ⊗ in . 󵄨 X ∂x n 󵄨󵄨󵄨γ(X)

66 | 3 Essentials of Non-Linear Elasticity Theory

Deformations of higher order. Since γ is smooth, one can use the Taylor formula [44]: γ(X + h) = γ(X) + γ󸀠 (X)[h] + ⋅ ⋅ ⋅ +

γ(r) (X)[h r ] + o(‖h‖r ) r!

as

h→0,

where the k-linear map³ γ(k) (X) ∈ Link (V, . . . , V; V) is a k-th order total derivative (k = 1, . . . , r) of γ at point X, and γ(k) (X)[h k ] := γ(k) (X)[h, . . . , h]. Thus, one has deformation gradients of high order: (r)

γ(X + h) = γ(X) + F X [h] + ⋅ ⋅ ⋅ +

F X [h r ] + o(‖h‖r ) r!

as

h→0,

(3.10)

(k)

where F X := γ(k) (X) for all k = 1, . . . , r. Referring again to the material fibers, one can say that deformed fiber γ(Y) − γ(X) with accuracy up to o(‖Y − X‖r ) is defined by the expression (r)

F X [Y − X] + ⋅ ⋅ ⋅ +

F X [Y − X, . . . , Y − X] . r!

The choice of r defines the desired accuracy and the corresponding theory.

3.3.2 Strain Measures Since V is Euclidean, one can obtain the expression for length of the deformed fiber. Let X ∈ SR be a point in the reference shape. For a point Y ∈ SR that lies near X, one has the expression γ(Y) − γ(X) = F X [Y − X] + o(‖Y − X‖) , for the deformed fiber, γ(Y) − γ(X). After multiplying this expression by itself, one arrives at ‖γ(Y) − γ(X)‖2 = F X [Y − X] ⋅ F X [Y − X] + o(‖Y − X‖2 )

as

Y→X.

Using the definition (12.3) of the transposed linear map, which in the case of F X takes the form ∀u, v ∈ V : (F X [u] ⋅ v = u ⋅ F TX [v]) , (3.11) where F TX ∈ Lin(V; V), one obtains the formula ‖γ(Y) − γ(X)‖2 = (Y − X) ⋅ F TX F X [Y − X] + o(‖Y − X‖2 ) as

3 k-linear maps are defined in Chapter 12, Section 12.5.2.

Y→X.

3.3 Deformation Gradient and Strain Measures

| 67

The tensor C X = F TX F X ∈ Sym(V) is called the right Cauchy–Green strain. Thus, the last equality may be written in the form ‖γ(Y) − γ(X)‖2 = (Y − X) ⋅ C X [Y − X] + o(‖Y − X‖2 )

Y→X.

as

(3.12)

With an accuracy up to o(‖Y − X‖2 ), the square of deformed fiber length is equal to (Y − X) ⋅ C X [Y − X]. Tensors C X , X ∈ SR define the smooth mapping (tensor field) C : SR → Sym(V) ,

X 󳨃→ C X ,

called the right Cauchy–Green tensor field (right Cauchy–Green strain). Consider the inverse deformation, γ−1 . Let x = γ(X). One has γ−1 (y) − γ−1 (x) = F −1 X [y − x] + o(‖y − x‖)

as

y→x.

Multiplying this relation by itself and denoting B X = F X F TX , one obtains 2 ‖γ−1 (y) − γ−1 (x)‖2 = (y − x) ⋅ B−1 X [y − x] + o(‖y − x‖ )

as

y→x.

(3.13)

With an accuracy up to o(‖y − x‖2 ), the square of reference fiber length is equal to T (y − x) ⋅ B −1 X [y − x]. The tensor B X = F X F X is called the left Cauchy–Green strain tensor. One has the field B : SR → Sym(V) , X 󳨃→ B X , called the left Cauchy–Green tensor field (left Cauchy–Green strain). Cauchy–Green measures define the following measures: C−1 (Finger strain) and −1 B (Almansi strain).

3.3.3 Deformation Gradient in Curvilinear Coordinates In the following paragraph, we obtain curvilinear coordinate representations using the results established in Section 3.2. Let γ : SR → S be a deformation. The general case. Let (Q i )3i=1 be curvilinear coordinates on SR and (q i )3i=1 be curvilinear coordinates on S. Using the relation (3.5) between the Cartesian representation γ d (3.1) of the deformation γ and the curvilinear representation γ c (3.2) of the same deformation γ, the chain rule and relations (3.3) and (3.4), from (3.7) one obtains FX =

󵄨 ∂q i 󵄨󵄨󵄨 󵄨󵄨 e i |γ(X) ⊗ E j |X . ∂Q j 󵄨󵄨󵄨σ̃ α (X)

(3.14)

Here, (∂q i )/(∂Q j ) are partial derivatives of γ c : (Q1 , Q2 , Q3 ) 󳨃→ (q1 , q2 , q3 ). One should note from the relation (3.14) that F X is not a tensor in the classical sense. It is

68 | 3 Essentials of Non-Linear Elasticity Theory

an example of a so-called two-point tensor, which is considered in detail in Chapter 5, Section 5.4. To obtain the relation for the transpose map F T let F = F i j e i ⊗ E j and F T = (F T )k l E k ⊗ e l . Applying u = E j , v = e l in (3.11), we obtain g il F i j = (F T )k l G jk . Multiplying both sides of this relation by G js and summing over j, we finally obtain (F T )s l = g il G js F i j .

(3.15)

The expression (3.14) can be represented in dyadic forms like e i ⊗ e j and E i ⊗ E j j as follows. One needs to use relations E i = Ω i e j and E i = ℧i j e j , established in Section 3.2. Thus, 󵄨 ∂q i 󵄨󵄨󵄨 j 󵄨󵄨 F X = ℧ k |X e i |γ(X) ⊗ e k |γ(X) , ∂Q j 󵄨󵄨󵄨σ̃ α (X) 󵄨 ∂q i 󵄨󵄨󵄨 󵄨󵄨 E k |X ⊗ E j |X . F X = ℧k i |X ∂Q j 󵄨󵄨󵄨σ̃ α (X) A particular case. If γ c = Id, then (∂q i )/(∂Q j ) = δ ij , and one has F i j = δ ij ,

(F T )s l = g il G is .

Since E i = G is E s and e i = g il e l , one obtains the following decompositions: F X = e i |γ(X) ⊗ E i |X ,

F TX = E i |X ⊗ e i |γ(X) .

3.3.4 Strain Measures in Curvilinear Coordinates We obtain the representation for the right Cauchy–Green strain in curvilinear coordinates. Since C = F T F, one has C = (F T )k i F i j E k ⊗ E j . This relation and (3.15) imply C X = g si |γ(X) G kl |X

󵄨 ∂q s 󵄨󵄨󵄨󵄨 ∂q i 󵄨󵄨󵄨 󵄨󵄨 E k |X ⊗ E j |X . 󵄨 ∂Q l 󵄨󵄨󵄨σ̃ α (X) ∂Q j 󵄨󵄨󵄨σ̃ α (X)

Analogously, one can obtain the representation for the left Cauchy–Green strain. Since B = FF T , one obtains B = (F T )k l F i k e i ⊗ e l . From the latter, it follows that B X = g sl |γ(X) G jk |X

󵄨 ∂q s 󵄨󵄨󵄨󵄨 ∂q i 󵄨󵄨󵄨 󵄨󵄨 e i |γ(X) ⊗ e l |γ(X) . 󵄨 󵄨 ∂Q j 󵄨󵄨σ̃ α (X) ∂Q k 󵄨󵄨󵄨σ̃ α (X)

Cauchy–Green strains are related with metrics by the following geometric relations. Applying the musical isomorphism (⋅)♭b , associated with the metric tensor G, to

3.4 Displacement Field |

69

C we obtain the field of bilinear forms C♭b , ♭

C Xb = g si |γ(X)

󵄨 ∂q s 󵄨󵄨󵄨󵄨 ∂q i 󵄨󵄨󵄨 󵄨󵄨 E m |X ⊗ E j |X . 󵄨󵄨 m j ∂Q 󵄨󵄨σ̃ α (X) ∂Q 󵄨󵄨󵄨σ̃ α (X)

The latter expression implies that C ♭b = γ ∗ g . Let (⋅)♯p denote the musical isomorphism associated with the metric tensor g. For the tensor B, we have B♯p = G jk |X

󵄨 ∂q m 󵄨󵄨󵄨󵄨 ∂q i 󵄨󵄨󵄨 󵄨󵄨 e i |γ(X) ⊗ e m |γ(X) . 󵄨 󵄨 ∂Q j 󵄨󵄨σ̃ α (X) ∂Q k 󵄨󵄨󵄨σ̃ α (X)

Thus,

∗

B♯p = γ∗ (G) .

3.4 Displacement Field For any point X ∈ SR and a deformation γ, a displacement vector is a vector u(X) := γ(X) − X. Thus, one has the smooth mapping u : SR → V ,

u : X 󳨃→ γ(X) − X ,

called a displacement vector field. The deformation gradient F is related with the displacement vector field u by the following relation: F X = u 󸀠 (X) + I , (3.16) where I ∈ Lin(V; V) is identity operator, i.e., I := IdV . The total derivative u 󸀠 (X) is called the displacement gradient, which is often denoted by ∇u(X) and may be used in the classical theory instead of⁴ F X . The right and left Cauchy–Green strain measures C X = F TX F X and B X = F X F TX can be represented in terms of displacements. Using the expression (3.16), one obtains C X = I + u󸀠 (X) + [u 󸀠 (X)]T + [u 󸀠 (X)]T u 󸀠 (X) , B X = I + u󸀠 (X) + [u 󸀠 (X)]T + u 󸀠 (X)[u 󸀠 (X)]T .

(3.17)

4 This is not true in the case of non-affine physical space. In the latter one does not have machinery to define difference between points.

70 | 3 Essentials of Non-Linear Elasticity Theory

3.5 Motion A motion is an one-parametric family of deformations: M = {γ t }t∈𝕋 , where 𝕋 ⊂ ℝ is an open interval, and γ t : SR → St is a deformation. Choosing a point X ∈ SR , one can define the curve χ M;X : 𝕋 → E , χ M;X (t) := γ t (X) , which represents a spatial trajectory of the point X. It is convenient to consider the motion M as the mapping γ : SR × 𝕋 → E ,

γ(X, t) := γ t (X) .

of two variables. Assume that for any X ∈ SR and t ∈ 𝕋 the partial mappings γ(⋅, t) : SR → E ,

γ(X, ⋅) : 𝕋 → E

are smooth. In this case, the motion M is called a smooth motion. For the smooth motion, one can define the deformation gradient F : SR × 𝕋 → Lin(V; V) ,

F(X, t) :=

∂γ(X, t) , ∂X

as well as the fields of velocity V and acceleration A, V : SR × 𝕋 → V , A : SR × 𝕋 → V ,

γ(X, t + s) − γ(X, t) , s V(X, t + s) − V(X, t) , A(X, t) := lim 𝕋t ∋s→0 s

V(X, t) := lim

𝕋t ∋s→0

where 𝕋t = {s ∈ ℝ | (s ≠ 0) ∧ (t + s ∈ 𝕋)}.

3.6 Compatibility Conditions The relation (3.9) shows that F can be considered as a triple of vector-valued 1-forms on SR . This observation allows one to involve such terms as exact form, closed form, etc., to study the compatibility question.

3.6.1 Review on de Rham Cohomology Let M be a smooth s-manifold. A smooth differential k-form ω ∈ Ω k (M) is closed, if dω = 0, and is exact, if ω = dη for some smooth differential (k − 1)-form η ∈ Ω k−1 (M). Since d ∘ d = 0 one has that each exact form is closed. The converse assertion is not

3.6 Compatibility Conditions | 71

true, in general. Its truth value depends on the topology of the underlying manifold and is studied in the framework of de Rham cohomology theory. Let 1 ≤ k ≤ s. Denote Z k (M) = ker(d : Ω k (M) → Ω k+1 (M)) , B k (M) = Im(d : Ω k−1 (M) → Ω k (M)) . The vector space Z k (M) is the space of all closed k-forms, and the vector space B k (M) is the vector space of all exact k-forms. Since B k (M) ⊂ Z k (M), one can introduce the following equivalence relation: ∀ω, η ∈ Ω k (M) :

(ω ∼ η) ⇔ (∃τ ∈ Ω k−1 (M) :

ω − η = dτ) .

k The corresponding quotient vector space is denoted by H dR (M) := Z k (M)/B k (M) and is referred to as the de Rham cohomology group in degree p of M [3, 61]. By the construction, the statement “each closed k-form on M is exact” is equivak lent to the equality H dR (M) = 0. In particular, one has (see [3], for instance): ∙ If M is a contractible smooth manifold, i.e., IdM is smoothly homotopic⁵ to a conk stant map, then H dR (M) = 0; ∙ If M is smooth manifold, then each point of M has a neighborhood in which every closed form is exact.

3.6.2 Necessary and Sufficient Conditions for Compatibility Compatibility and exactness. Let γ : SR → S be a deformation. Since each component mapping γ i , i = 1, 2, 3, is a smooth real-valued function, one can apply the exterior differential to it. This implies dγ i = F i ∈ Ω1 (SR ) ,

i = 1, 2, 3 .

So, the obtained F i are exact forms on SR . Conversely, if F i ∈ Ω1 (SR ), i = 1, 2, 3, are some exact 1-forms, then one can find three smooth functions γ i : SR → ℝ, such that F i = dγ i . These functions can be considered as component functions of some smooth mapping γ. For this mapping, Tγ = ∂ x i ⊗ F i . Let F i ∈ Ω1 (SR ), i = 1, 2, 3, be 1-forms. If there exists a smooth mapping γ : SR → E, for which F i = dγ i , then these forms are called compatible. One obtains the following necessary and sufficient condition: 1-forms F i ∈ Ω1 (SR ), i = 1, 2, 3, are compatible iff they all are exact.

5 Two smooth mappings f0 and f1 between smooth manifolds M and N are smoothly homotopic if there exists a smooth mapping F : M × [0, 1] → N, such that F(⋅, i) = f i , i = 0, 1.

72 | 3 Essentials of Non-Linear Elasticity Theory

Compatibility and contractibility. The question of exactness is intimately related with the inner properties of the topological space SR . Suppose that SR is a contractible space, i.e.; intuitively, it can be continuously shrunk to a point. Then, H 1dR (SR ) = 0 and the exactness can be replaced by the requirement of closeness: 1-forms F i , i = 1, 2, 3 are compatible iff they all are closed. Thus, one needs to check that dF i = 0, i = 1, 2, 3. In classical mechanics, this condition is written as curl F = 0 or, using ∇, as ∇ × F = 0. Compatibility in the general case. The conditions dF i = 0, i = 1, 2, 3 only become necessary and lose their sufficiency status when SR has “holes” and other pathologies that break its contractibility. The following theorem defines the conditions of compatibility. Claim. The necessary and sufficient conditions for compatibility of F i are: (i) dF i = 0, i = 1, 2, 3; (ii) ∫c F i = 0, i = 1, 2, 3, k = 1, . . . , β 1 (SR ). Here, β 1 (SR ) is a Betti number, calcuk lated for SR , and c k are generators of⁶ H1 (SR ; ℝ). Proof of this claim is given in [63]. Remark 3.2. A Betti number β k (SR ) is a topological invariant that refers to the number of k-dimensional holes. Let us briefly give the formal description of Betti numbers [62, 64]. Let s ∈ ℕ, then denote by ∆ s ⊂ ℝs the standard s-simplex, formed by vectors s (e i )i=0 , where e0 = 0 and for i ∈ {1, . . . , s}, e i = (0, . . . , 1, . . . 0), in which 1 stands in the sth place. That is, ∆ s has vertices in the zero vector and in vectors of the standard basis. Let M be a topological space. A continuous map σ : ∆ s → M is called a singular s-simplex in M. If s = 0, then the image, σ(∆0 ), is just a point in M. If s = 1, then ∆1 = [0, 1] is a closed interval, and σ is a path in M. The set S s (M) of all singular s-simplices in M can serve as a base for constructing a group. Let σ ∈ S s (M). Then, one can generate a group F(σ) as follows: F(σ) = {σ} × ℤ, and the multiplication operation is defined as (σ, l)(σ, k) := (σ, l + k). Thus, we arrive at the sequence (F(σ))σ∈S s (M) of groups. Any finite tuples (g1 , . . . , g l ) and (h1 , . . . , h k ), where g i , h j belong to some F(σ), can be multiplied just by concatenation: (g1 , . . . , g l )(h1 , . . . , h k ) := (g1 , . . . , g l , h1 , . . . , h k ). This operation is associative, and the sequence of the length zero is its identity element. However, it does not satisfy the condition of inverses. Such an issue can be solved (see [64] for details) and we arrive, finally, at the group C s (M) on the set S s (M). This group is called the singular chain group in dimension s. Any of its elements that can be written as a formal linear combination of singular simplices with integer coefficients are called a singular s-chain in M.

6 Here H 1 (SR ; ℝ) is first homology group with real coefficients [62].

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Let us comment on the intuitive view on singular chains. If chain “closes up” on itself and is not a “boundary” of any chain with higher dimension, then it must surround a “hole”. Thus, in the framework of this view, a chain is an object that allows detection of “holes”. The boundary of a chain is composed of faces of simplices that constitute the chain. From a formal point of view, for each k = 0, 1, . . . , s, we define an affine map F k,s : ∆ s−1 → ∆ s , which works as follows: {e i , i ∈ {0, . . . , i − 1} , F k,s (e i ) := { e , i ∈ {i, . . . , s − 1} . { i+1 By the definition, F k,s maps ∆ s−1 in the boundary face of ∆ s that is situated opposite to the vertex e i . Now, let σ : ∆ s → M be a singular simplex, which defines a (s − 1)-chain ∂ p σ by the relation s

∂ s σ := ∑ (−1)k σ ∘ F k,s . k=0

The chain ∂ s σ is called the boundary of σ, and the homomorphism ∂ s : C s (M) → C s−1 (M) is the boundary operator. An s-chain c is called a cycle if ∂ s c = 0 and a boundary if c = ∂ s+1 c1 for some (s + 1) chain c1 . Consider an example. If σ : ∆1 → M, then ∂σ = σ(1) − σ(0). If σ is a cycle, then σ(0) = σ(1), and σ is a closed path. Denote Z s (M) = ker ∂ s , that is, a group of all s-cycles, and B s (M) = ∂ s+1 (C s+1 (M)), that is, a group of images. Both Z s (M) and B s (M) are subgroups of C s (M). The operation ∂ s satisfies the property ∂ s ∘ ∂ s+1 = 0. Thus, B s (M) is a subgroup of Z s (M). We arrive at the following definition; the sth singular homology group of M is the quotient group H s (M) := Z s (M)/B s (M) . Appealing again to intuitive reasonings, one should say that H s (M) = 0 iff no “sdimensional holes”. That is, every s-cycle is a boundary of some (s + 1)-chain. Finally, the integer β s (M) := rank(H s (M)) is called the sth Betti number of M. In this definition, the item rank(H s (M)) is a number of elements in any finite basis for H s (M). That is, any element of H s (M) can be decomposed on a formal linear combination of β s (M) elements. Local compatibility conditions. The physical space that contains shapes is Euclidean. Thus, its Riemannian curvature tensor vanishes, R(g) = 0, where g is a metric tensor on E. The reference shape is also Euclidean. Thus, the pullback of the curvature vanishes as well: γ∗ R(g) = 0. One has γ∗ R(g) = R(γ∗ g) = R(C♭ ) , where C♭ is a field of bilinear forms obtained by lowering index of C. Therefore, a necessary condition for compatibility of C is R(C♭ ) = 0. As was shown in [60], this condition is locally sufficient as well.

74 | 3 Essentials of Non-Linear Elasticity Theory Remark 3.3. Applying the musical isomorphism (⋅)♭ to the “first leg”, we arrive at the following representation: C♭X = C mn |X i m ⊗ i n ,

3

where

C mn |X = ∑ k=1

󵄨 󵄨 ∂x k 󵄨󵄨󵄨 ∂x k 󵄨󵄨󵄨 󵄨 󵄨󵄨 . 󵄨 ∂X m 󵄨󵄨󵄨X ∂X n 󵄨󵄨󵄨X

In componentwise form, the equality R(C♭ ) = 0 can be written in Cartesian coordinates as R(C♭ ) =

1 (∂ k ∂ j C im + ∂ i ∂ m C jk − ∂ m ∂ j C ik − ∂ i ∂ k C jm ) + (C−1 )rs (A jkr A ims − A jmr A iks ) = 0 , 2 (3.18)

where A ijk = A jik =

1 (∂ j C ik + ∂ i C jk − ∂ k C ij ) . 2

In infinitesimal strain approximation, the quadric terms in (3.18) are neglected, and this condition can be written in more simple form, namely the Saint-Venant equation: ∂ k ∂ j ε im + ∂ i ∂ m ε jk − ∂ m ∂ j ε ik − ∂ i ∂ k ε jm = 0 . The version of this equation in curvilinear coordinates can be found in [65]. Using ∇, one can write Ink(ε) = 0 ,

Ink(ε) = curl curlT ε = ∇ × (∇ × ε)T .

Here, ε is an infinitesimal strain, ε := [∇u]sym , and u is a displacement vector field. In classical works on conventional elasticity, the integrability conditions are formulated in terms of fields of infinitesimal strains, ε = [∇u]sym , and vortices, ω = [∇u]asym , in the following manner: → 󳨀 ∇ × ε − (∇ ω)T = 0 , → 󳨀 where ω is a vector field, associated with ω. Raising the differentiation order in the left part of this equality, one obtains independent conditions for ε and ω. These are the Saint-Venant equation ∇ × (∇ × ε)T = 0 , and

→ 󳨀 ∇⋅ω.

At the same time, these conditions are necessary, but insufficient; when these conditions are satisfied, the displacement field corresponding to the given ε and ω is found up to a rigid motion and the gradient of some scalar function [66]. If this arbitrariness is leveled by additional conditions (for example, boundary conditions), then the displacements are uniquely determined by Cesaro’s formulae. In terms of the bend-twist

3.7 Stresses

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→ 󳨀 tensor⁷ κ = ∇ω one can formulate more general compatibility conditions [66]: κ = −(∇ × ε)T .

∇×κ =0,

(3.19)

These conditions are necessary and sufficient. The latter condition explicitly shows the relation between infinitesimal strains and vortices. Remark 3.4. It is possible to derive the compatibility conditions without reasoning from non-Euclidean geometry. In the case of infinitesimal strains, it is a trivial consequence of the symmetry of partial derivatives with respect to the order of differentiation (Schwartz’s theorem). It is reproduced in all textbooks related to linear elasticity. For finite strains, this result may be obtained by direct differentiation of the Euclidean strains, see, for example, [68]. Displacements in the reference shape with a given strain tensor. For given 1-forms F i , we can find the positions of reference points in the actual shape: p

x(p) = ∫ ∂ i ⊗ F i ⌞ χ̇ , p0

where χ is a curve on reference shape that starts in p0 and ends in p. If fields κ and ε are known and satisfy (3.19), then the sought vortices and displacements are determined by the Cesaro-type formulae [66]: r

→ 󳨀 → 󳨀 ω(r) = ω(r 0 ) + ∫ dr 󸀠 ⋅ κ(r 󸀠 ) , r0 r

→ 󳨀 u(r) = u(r 0 ) + ω(r 0 ) × (r − r 0 ) + ∫ dr 󸀠 ⋅ {ε(r 󸀠 ) + κ(r 󸀠 ) × (r − r 󸀠 )} , r0

where r 0 is the position vector of a fixed point. These relations determine single-valued fields that do not depend on the path of integration, as it follows from the Stokes’ theorem and the compatibility conditions (3.19).

3.7 Stresses Ideas that led to the mathematical formulation of stresses⁸ are rooted in works of James Bernoulli, John Bernoulli, and L. Euler [71, 72]. The concept of stresses as a tensor field of the second rank, generalizing the concept of hydrostatic pressure in an ideal fluid,

7 In [67] the tensor κ is called a curvature-twist dyadic. 8 A detailed exposition of the historical aspects of the theory of stresses is given in [69, 70].

76 | 3 Essentials of Non-Linear Elasticity Theory

was formulated in the works of Cauchy [73, 74] and developed in the studies of Poisson and Navier [75]. Note that the coordinate representation of such fields assumed a certain analytical freedom, since different coordinate systems associated with the reference and deformed shapes are used. In this case, there is a need for coordinate transformations, the general form of which was given by Piola [76]. The classical Cauchy theory is based on the cut and freezing principles⁹. According to the first one, when a part U is cut from the actual shape S with an arbitrary piecewise smooth boundary ∂U, the action of the remaining part of this shape on U can be replaced by the contact force density t n distributed over ∂U [78, 79]. Here, n is an outward-pointing unit normal vector field on ∂U. It is postulated that the resultant force FU , acting on the part U, is represented as the sum of the contact and body forces: FU = ∫ t n dS + ∫ b dV , ∂U

U

where b is the body force density. To formalize the contact forces, Cauchy used the postulate on the dependence of their density t n only on the normal n to the cut surface, i.e., t n (x) := t(x, n) . This allowed him (under certain agreements on smoothness and using the famous tetrahedron argument) to prove the “action and reaction equality”: −t n = t −n , and to prove the existence of stress tensor field T : S → V ⊗ V ∗ , i.e., the field of linear transformation of the outer unit normal n to the contact force density t n : t n = Tn . Remark 3.5. In the middle of the twentieth century, the Cauchy theorem on the existence of a stress tensor was reformulated in Noll’s works, and a more general proof was constructed, which used not three linearly independent vectors, but only two. The proof was divided into a proof of the homogeneity and additivity of the relations determining the relationship between the unit normal and the density of contact forces [80]. Note that in classical reasonings, the part of the shape was considered as a “nice” subset of Euclidean space. Later, in the framework of potential theory (O. Kellogg [58]), the notion of “nice” subset was clarified: it is open set that is bounded by a finite

9 It seems that the cut principle dates back to the works of Galileo, and the freezing principle was formulated by Simon Stevin [77] in 1586 (Stevin S. “De beghinselen des waterwichts”, 1586). A copy of the corresponding page can be found in [70, p. 198]. Note that Stevin considered the equilibrium of a liquid, mentally transferring the equilibrium laws for a rigid body to an arbitrary part of the liquid. This is reflected in the term “freezing”.

3.7 Stresses

| 77

number of smooth regular elementary surfaces. Each of them can be represented as a graph of some smooth function (see Section 3.1). This requirement allows one to use Stokes’ theorem in its Euclidean formulation. Then, applying the freezing principle, using Eulerian Dynamics laws, one can write ∫ t n dS + ∫ b dV = ∂U

U

d ∫ ρ u̇ dV , dt U

∫ p × t n dS + ∫ p × b dV = U

∂U

d ∫ ρp × u̇ dV . dt U

Here, u is a displacement vector field, p is a position vector field, and ρ is the mass density. Since U is arbitrary, one obtains the differential equations of motion: ∇ ⋅ T − ρ ü + b = 0 ,

TT = T .

Suppose that we are given reference, SR , and actual, S, shapes that are related by a deformation γ : SR → S. Let U0 be a part of SR , which is a shape itself, and U = γ(U0 ). Let N, n be outward-pointing unit normal vector fields on ∂U0 and ∂U. The contact force density t n and the Cauchy stress tensor T : S → V ⊗ V ∗ take part in the spatial description, i.e., they allow one to calculate contact force acting on ∂U in actual coordinates. Transforming the surface integral over ∂U into an integral over ∂U0 , one obtains [59] C = ∫ Tn dS = ∫ PN dS0 , ∂U

∂U0

where dS0 is the reference area element, dS is the actual area element related by Nanson’s formula [8] n dS = JF −T NdS0 , and P : SR → V ⊗ V ∗ is the first Piola–Kirchhoff stress tensor related with T by Piola transformation [37] JT = PF T , where J is the Jacobian of F (3.8). Introducing the contact force density t R;N in the reference coordinates, one arrives at t R;N = PN . Remark 3.6. In the framework of this book, we use Cauchy and first Piola–Kirchhoff stress tensors, although other stress measures exist. The second Piola–Kirchhoff stress tensor S and the Kirchhoff stress tensor τ may serve as examples. They are related with Cauchy and first Piola–Kirchhoff stress tensors as follows [81]: S = F −1 P ,

τ = JT .

78 | 3 Essentials of Non-Linear Elasticity Theory

3.8 Non-Linear Elasticity as Field Theory 3.8.1 Action and Its Lagrangian In this section, we consider the elements of field theory that allow us to derive equations of motion and conservation laws in spatial and material descriptions by means of the most “economical” reasoning (E. Mach), provided by variational calculus. Let SR be any shape taken as a reference and M ⊂ ESR ×𝕋 be the set of all motions. The main object of consideration is represented by the action A, which is a mapping A: M → ℝ . We apply the principle of additivity of action: there exists the smooth mapping L : SR × 𝕋 × E × V × Lin(V; V) × ⋅ ⋅ ⋅ × Linr (V, . . . , V; V) → ℝ , L : (X, t, x, u, f 1 , . . . , f r ) 󳨃→ L(X, t; x, u, f 1 , . . . , f r ) , called Lagrangian volume density, such that t2

A(γ) = ∫ ∫ L(X, t, γ(X, t), V (X, t), F(X, t), . . . , F (r) (X, t)) dV dt ,

(3.20)

t1 R

for any γ ∈ M, t1 , t2 ∈ 𝕋, and R ⊂ SR , which is a shape itself. The value of r in (3.20) represents the order of gradient theory. Remark 3.7. The list of variables presented in Lagrangian density depends on the theory considered. For example, with thermoelasticity, one should add a scalar field of temperature to the list. In the more general case, the Lagrangian density may also contain mixed derivatives with respect to space and time and high derivatives with respect to time. Further, we consider the case when r = 1. That is, formula (3.20) reduces to t2

A(γ) = ∫ ∫ L(X, t, γ(X, t), V (X, t), F(X, t)) dV dt .

(3.21)

t1 R

We say that the theory considered is the first-order gradient theory. Moreover, we restrict ourselves to the following form of Lagrangian density¹⁰: L(X, t, γ(X), V (X), F(X)) =

1 ρ R (X)V (X)2 − W(X, F(X)) − Φ(γ(X), t) , 2

(3.22)

where ρ R is the volume density of the solid, relative to the reference configuration, W is the elastic energy volume density, and Φ is the potential of physical body forces.

10 Here symbol X stands for (X, t).

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| 79

Remark 3.8. Equivalently, the action (3.21) may be written in the form t2

A(γ) = ∫ (K − W + P) dt , t1

where K = 12 ∫R ρV 2 dV is the kinetic energy, W = ∫R W dV is the stored elastic (potential) energy, and P = − ∫R Φ dV is work, generated by physical body forces.

3.8.2 Partial and Full Variations We introduce a one-parameter family {g(ε)} ε∈ℝ of transformations of both coordinates and fields. Elements of this family constitute a Lie group G (Chapter 14) with the group properties g(0) = Id , g(ε) ∘ g(μ) = g(ε + μ) . The action (X, t, γ(X, t)) 󳨃→ g(ε) ∘ (X, t, γ(X, t)) of G on a triple (X, t, γ(X, t)) generates the following mappings: ̃ : (X, t, ε) 󳨃→ X(X, ̃ X t, ε) , ̃t : (X, t, ε) 󳨃→ ̃t(X, t, ε) , ̃ ̃γ : (X, t, ε) 󳨃→ ̃γ(X(X, t, ε), ̃t(X, t, ε); ε) . ̃ ̃ Here, X(X, t, 0) = X, ̃t(X, t, 0) = t, and ̃γ(X(X, t, 0), ̃t(X, t, 0); 0) = γ(X, t). Under action of the group G, the integral (3.21) transforms into ̃t 2 (ε)

̃ A(ε) = ∫

̃ ̃ d̃t(ε) . ∫ L(X(X, t, ε), ̃t(X, t, ε), . . . ) d V(ε)

(3.23)

̃t 1 (ε) R(ε) ̃

Variations of spatial and temporal variables have the form¹¹: ̃ 󵄨󵄨󵄨 󳨀→ d X 󵄨󵄨 δε , δX = 󵄨 dε 󵄨󵄨󵄨ε=0

δt =

󵄨 d̃t 󵄨󵄨󵄨 󵄨󵄨 δε . dε 󵄨󵄨󵄨ε=0

By definition, full variation of the deformation can be expressed as δγ = and, consequently,

̃ ∂ ̃γ d̃t ∂ ̃γ 󵄨󵄨󵄨 ∂ ̃γ d X d ̃γ 󵄨󵄨󵄨󵄨 + + )󵄨󵄨 δε , 󵄨󵄨 δε = ( ̃ dε ∂̃t dε ∂ε 󵄨󵄨󵄨󵄨ε=0 dε 󵄨󵄨ε=0 ∂X 󳨀→ δγ = δγ + F[δX] + V δt ,

󳨀→ 11 Note that δX is a vector field and δt is a scalar field.

(3.24)

80 | 3 Essentials of Non-Linear Elasticity Theory

where δγ denote partial variation δγ =

∂ ̃γ 󵄨󵄨󵄨󵄨 󵄨 δε . ∂ε 󵄨󵄨󵄨ε=0

We introduce the following mappings: f : SR × 𝕋 → SR × 𝕋 × E × V × Lin(V; V) , such that f(X, t) := (X, t, γ(X, t), V (X, t), F(X, t)), and ̃f : SR × 𝕋 × ℝ → SR × 𝕋 × E × V × Lin(V; V) , ̃ such that f(X, t, ε) := L(X(X, t, ε), ̃t(X, t, ε), . . . ). Compositions of these maps with the Lagrangian density, L : (X, t, x, u, f ) 󳨃→ L(X, t, x, u, f ) lead to the following expressions: ̂ = L ∘ f : (X, t) 󳨃→ L(X, t, γ(X, t), V (X, t), F(X, t)) , L ̃ = L ∘ ̃f : (X, t, ε) 󳨃→ L(X(X, ̃ L t, ε), ̃t(X, t, ε), . . . ) . In the present section we will use the following notation. If h : SR × 𝕋 × E × V × Lin(V; V) → W is any mapping with values in some set W, then h|f denotes h ∘ f. This is a mapping defined on SR × 𝕋 with values in W. A similar notation for ̃f is used. First form for δA. Now we consider the variation of the action (3.23). Taking into account that integration region depends on ε, one arrives at the formula 󵄨󵄨 ̃t 2 (ε) 󵄨󵄨 ̃ 󵄨󵄨󵄨 ̃ ̃ ̃ 󵄨󵄨 dA d L d X d d t 󵄨󵄨 δε = [ ̃ Div ( ̃ ̃ d̃t ] δA = +L )+L ) dV [ ∫ ∫ ( ]󵄨󵄨󵄨 δε , 󵄨󵄨 󵄨󵄨 dε 󵄨󵄨ε=0 dε dε dt dε 󵄨 ̃ [̃t 1 (ε) R(ε) ]󵄨󵄨ε=0 or, equivalently, t2

→ ̂ d ̂ +L ̂ divR 󳨀 δt) dV dt . δX + L δA = ∫ ∫ (δL dt

(3.25)

t1 R

̃ 󵄨󵄨 ̂ = dL 󵄨󵄨󵄨󵄨 δε is a variation of the Lagrangian density that can be calculated Here, δL dε 󵄨󵄨󵄨ε=0 by the chain rule as follows: ̂=( δL

̃ ∂L 󵄨󵄨󵄨󵄨 d X ∂L 󵄨󵄨󵄨󵄨 d̃t ] + 󵄨󵄨 [ 󵄨 [ ]+ ∂X 󵄨󵄨̃f dε ∂t 󵄨󵄨󵄨̃f dε ∂L 󵄨󵄨󵄨󵄨 d ̃γ̇ + 󵄨 [ ]+ ∂u 󵄨󵄨󵄨̃f dε

∂L 󵄨󵄨󵄨󵄨 d ̃γ 󵄨 [ ] ∂x 󵄨󵄨󵄨̃f dε 󵄨 ∂L 󵄨󵄨󵄨󵄨 d ∂ ̃γ 󵄨󵄨󵄨󵄨 ]) [ 󵄨 δε . 󵄨 ̃ 󵄨󵄨󵄨 ∂f 󵄨󵄨󵄨̃f dε ∂ X 󵄨ε=0

3.8 Non-Linear Elasticity as Field Theory |

81

Taking into account the equalities ̂ ∂L ∂L 󵄨󵄨󵄨󵄨 = 󵄨 + ∂X ∂X 󵄨󵄨󵄨f ̂ ∂L 󵄨󵄨󵄨󵄨 ∂L = 󵄨 + ∂t ∂t 󵄨󵄨󵄨f

∂L 󵄨󵄨󵄨󵄨 󵄨 ∂x 󵄨󵄨󵄨f ∂L 󵄨󵄨󵄨󵄨 󵄨 ∂x 󵄨󵄨󵄨f

∂γ ∂L 󵄨󵄨󵄨󵄨 ∂ γ̇ ∂L 󵄨󵄨󵄨󵄨 ∂F + + , 󵄨󵄨 󵄨 ∂X ∂u 󵄨󵄨f ∂X ∂f 󵄨󵄨󵄨f ∂X ∂γ ∂L 󵄨󵄨󵄨󵄨 ∂ γ̇ ∂L 󵄨󵄨󵄨󵄨 ∂F + + , 󵄨 󵄨 ∂t ∂u 󵄨󵄨󵄨f ∂t ∂f 󵄨󵄨󵄨f ∂t

(3.26) (3.27)

together with (3.24), we arrive at the following result¹²: ̂= δL

̂ ̂ 󳨀→ ∂L 󵄨󵄨󵄨󵄨 d ∂L 󵄨󵄨󵄨󵄨 ∂δγ ∂L ∂L 󵄨󵄨󵄨󵄨 ∂L [δX] + ] . [δt] + 󵄨󵄨 [δγ] + 󵄨󵄨 [ δγ] + 󵄨 [ ∂X ∂t ∂x 󵄨󵄨f ∂u 󵄨󵄨f dt ∂f 󵄨󵄨󵄨f ∂X

(3.28)

Values of x 󳨃→ ∂L (x) are linear mappings from Lin(V; V) to ℝ. Since the space ∂f Lin(V; V) is endowed with Euclidean structure with scalar product (:), there exists a unique field Lf : x 󳨃→ Lf (x) ∈ Lin(V; V) of linear mappings, such that for any field T : x 󳨃→ T(x) ∈ Lin(V; V), the following equality holds: ∂L [T] = Lf :T . ∂f ∂L Similar representations hold for fields ∂L ∂x and ∂u of linear functionals. There exists a unique vector field Lx : x 󳨃→ Lx (x) ∈ V and Lu : x 󳨃→ Lu (x) ∈ V, such that for any vector field v : x 󳨃→ v(x) ∈ V, the following representations hold:

∂L [v] = Lx ⋅ v , ∂x

and

∂L [v] = Lu ⋅ v . ∂u

̂ takes the form: Hence, the expression (3.28) for δL ̂= δL

̂ ̂ 󳨀→ d ∂L ∂L 󵄨 ∂δγ [δX] + . [δt] + Lx |f ⋅ δγ + Lu |f ⋅ δγ + Lf 󵄨󵄨󵄨f : ∂X ∂t dt ∂X

(3.29)

Substituting (3.29) into (3.25) taking into account (2.12) and (2.13), and the equality d d d ( Lu |f ⋅ δγ) = Lu |f ⋅ δγ + Lu |f ⋅ δγ , dt dt dt one can obtain t2

→ d ̂ 󵄨 ̂󳨀 δX + LTf 󵄨󵄨󵄨󵄨 [δγ]) + (Lδt + Lu |f ⋅ δγ) δA = ∫ ∫ {divR (L f dt t1 R

d 󵄨 ( Lu |f )) ⋅ δγ} dV dt . + ( Lx |f − divR ( Lf 󵄨󵄨󵄨f ) − dt

(3.30)

12 Note that partial variation commutes with differential operators. That is, δ γ̇ = (d/dt)δγ, and ∂γ ∂δγ δ = . ∂X ∂X

82 | 3 Essentials of Non-Linear Elasticity Theory

Using the following notations for Noether current components 󳨀→ 󵄨 δJ 1 = L|f δX + LTf 󵄨󵄨󵄨󵄨 [δγ] , f δJ 2 = L|f δt + Lu |f ⋅ δγ ,

(3.31) (3.32)

and the Euler–Lagrange operator d 󵄨 𝔼γ = Lx |f − divR ( Lf 󵄨󵄨󵄨f ) − ( L u |f ) , dt

(3.33)

one can rewrite (3.30) more concisely: t2

δA = ∫ ∫ {divR δJ 1 + t1 R

d δJ 2 + 𝔼γ ⋅ δγ} dV dt . dt

(3.34)

Second form for δA. Now we obtain the second form for δA. Keeping in mind the expressions (3.26), (3.27), and (3.29), as well as the relation (3.24) between partial and full variations, one can derive the equality → ̂ d ̂ +L ̂ divR 󳨀 δt δL δX + L dt 󳨀→ ̂ d ∂L 󵄨󵄨󵄨󵄨 󳨀→ ∂L 󵄨󵄨󵄨󵄨 ∂γ ̂ 󵄨 = + L divR δX + L δt . 󵄨󵄨 [δX] + 󵄨 δt + Lx |f ⋅ δγ + Lu |f ⋅ δ γ̇ + Lf 󵄨󵄨󵄨f :δ ∂X 󵄨󵄨f ∂t 󵄨󵄨󵄨f ∂X dt Taking into account commutation relations d d ∂γ d 󳨀→ d γ= δγ − [ δX] − γ̇ δt , dt dt ∂X dt dt 󳨀→ ∂ ∂γ ∂ δX ∂ ∂ γ= δγ − − γ̇ ⊗ δt , δ ∂X ∂X ∂X ∂X ∂X δ

and the following identities: 󳨀→ 󳨀→ ∂ δX divR δX = I: , ∂X 󳨀→ 󳨀→ ∂γ T 󵄨󵄨 ∂ δX 󵄨󵄨 ∂γ ∂ δX =( ) Lf 󵄨󵄨f : , Lf 󵄨󵄨f : ∂X ∂X ∂X ∂X ∂ ∂ 󵄨 󵄨 δt) = LTf 󵄨󵄨󵄨󵄨 [γ]̇ ⌟ δt , Lf 󵄨󵄨󵄨f : (γ̇ ⊗ f ∂X ∂X one can obtain the following expression for δA: t2

δA = ∫ ∫ { t1 R

∂L 󵄨󵄨󵄨󵄨 󳨀→ ∂ 󳨀→ ∂L 󵄨󵄨󵄨󵄨 ∂γ T 󵄨󵄨 δX ) Lf 󵄨󵄨f ) : 󵄨󵄨 [δX] + 󵄨󵄨 δt + ( L|f I − ( 󵄨 󵄨 ∂X 󵄨f ∂t 󵄨f ∂X ∂X

d d δt + Lx |f ⋅ δγ + Lu |f ⋅ δγ dt dt d 󳨀→ ∂ ∂γ T 󵄨 󵄨 ∂ −( ) [ Lu |f ] ⋅ δX + Lf 󵄨󵄨󵄨f : δγ − LTf 󵄨󵄨󵄨󵄨 [γ]̇ ⌟ δt} dV dt . f ∂X dt ∂X ∂X

+ ( L|f − Lu |f ⋅ γ)̇

(3.35)

3.8 Non-Linear Elasticity as Field Theory | 83

Standard notations. Now introduce notations for the fields: ∙ Eshelby energy-momentum tensor e = L|f I − ( ∙

∂γ T 󵄨󵄨 ) Lf 󵄨󵄨f , ∂X

Hamiltonian density H = L|f − Lu |f ⋅ γ̇ ,

∙

Energy flux (Umov–Poynting vector) 󵄨 U = LTf 󵄨󵄨󵄨󵄨 [γ]̇ , f

∙

Density of the canonical impulse K=(

∙

∂γ T ) [ L u |f ] , ∂X

Density of the physical impulse p = L u |f ,

∙

The first Piola–Kirchhoff stress tensor 󵄨 P = − Lf 󵄨󵄨󵄨f .

With such notations, the expression (3.35) may be rewritten in the form: t2

δA = ∫ ∫ { t1 R

∂L 󵄨󵄨󵄨󵄨 󳨀→ ∂L 󵄨󵄨󵄨󵄨 ∂ 󳨀→ d δX + H δt + Lx |f ⋅ δγ 󵄨󵄨 [δX] + 󵄨 δt + e: ∂X 󵄨󵄨f ∂t 󵄨󵄨󵄨f ∂X dt

+ p⋅

d d 󳨀→ ∂ ∂ δγ − K ⋅ δX − P: δγ − U ⌟ δt} dV dt . dt dt ∂X ∂X

(3.36)

Remark 3.9. Customarily, the following agreements on notations are used in continuum mechanics. Variables of the Lagrangian density L are denoted by the same symbols as the corresponding fields. That is, one writes L : (X, t, γ, V , F) 󳨃→ L(X, t, γ, V , F) , ∂L ∂L ∂L ∂L ∂L and use notations ∂L ∂γ , ∂V , ∂F for ∂x , ∂u , ∂f . Although these notations are not strictly correct, they are convenient. Another customarily (but also strictly not correct) agreement is that Lf is denoted as ∂L ∂F . Thus, one have the same notation for the 3rd order tensor field and for the corresponding a unique 2nd order tensor field which is associated with the former by the operation (:). Similar remark holds for Lx and Lu . By virtue of such agreements (3.33) has the following representation:

𝔼γ =

∂L d ∂L ∂L − divR − . ∂γ ∂F dt ∂V

84 | 3 Essentials of Non-Linear Elasticity Theory

3.8.3 Field Equations Consider the particular case when only physical fields (i.e., γ) undergo variations; then formula (3.34) reduces to t2

δA = ∫ ∫ 𝔼γ δγ dV dt . t1 R

From this follows the field equation d 󵄨 Lx |f − divR ( Lf 󵄨󵄨󵄨f ) − ( L u |f ) = 0 , dt or, with the notation introduced above, Lx |f + divR P =

d p. dt

Here, Lx |f denotes the mass forces. Since L is defined by (3.22), we arrive at the following field equation in the direct motion description¹³: ∂W ∂Φ divR (3.37) − = ρ R V̇ . ∂F ∂γ

3.8.4 Action Invariance Conditions Consider another particular case, when only coordinates undergo variations. Moreover, we suppose that these variations are defined by a Lie group. ̃ I. Spatial translations. Let X(X, t, ε) = X + εX 0 , where X 0 ∈ V is a constant vector, and let ̃t(X, t, ε) = t, ̃γ(X, t, ε) = γ. Since 󳨀→ δX = X 0 δε ,

d 󳨀→ δX = 0 , dt

∂ 󳨀→ δX = 0 , ∂X

the variation (3.36) reduces to t2

δA = ∫ ∫ { t1 R

∂L 󵄨󵄨󵄨󵄨 󵄨 [X 0 ]δε} dV dt . ∂X 󵄨󵄨󵄨f

The equality δA = 0 must be valid for each spatial translation. Since X 0 is arbitrary, we arrive at the following condition: ∂L 󵄨󵄨󵄨󵄨 󵄨 =0. ∂X 󵄨󵄨󵄨f ∂W denote the third-order tensor field. However, traditionally this symbol is ∂F used for a unique second-order tensor field, which is associated with the former by the operation (:). We mean the latter case.

13 Note that formally

3.8 Non-Linear Elasticity as Field Theory | 85

II. Temporal translations. Let ̃t(X, t, ε) = t + εt0 , where t0 ∈ ℝ is a constant, and let ̃ X(X, t, ε) = X, ̃γ(X, t, ε) = γ. Since δt = t0 δε ,

d δt = 0 , dt

∂ δt = δt , ∂X

the variation (3.36) reduces to t2

δA = ∫ ∫ { t1 R

∂L 󵄨󵄨󵄨󵄨 󵄨 t0 δε} dV dt . ∂t 󵄨󵄨󵄨f

The equality δA = 0 must be valid for each temporal translation. Since t0 is arbitrary, we arrive at the following condition: ∂L 󵄨󵄨󵄨󵄨 󵄨 =0. ∂t 󵄨󵄨󵄨f ̃ III. Spatial rotations. Let X(X, t, ε) = o + Q(ε)[X − o], where o ∈ E is a fixed point and ℝ ∋ ε 󳨃→ Q(ε) ∈ O(3) is smooth field of orthogonal mappings, such that Q(0) = I. Temporal coordinate and fields are fixed: ̃t(X, t, ε) = t, ̃γ(X, t, ε) = γ. Since Q is smooth, we may write Q(ε) = Q(0) + εω + o(ε) where ω =

as

ε→0,

dQ(ε) 󵄨󵄨󵄨󵄨 󵄨 [1] ∈ Lin(V; V). dε 󵄨󵄨󵄨ε=0

Remark 3.10. Linear mapping ω is antisymmetric i.e., ωT = −ω. This expression may be obtained in the following way. One should differentiate the identity Q(ε)Q T (ε) = I with respect to ε and put ε = 0. According to the expressions 󳨀→ δX = ω[X − o]δε ,

d 󳨀→ δX = 0 , dt

∂ 󳨀→ δX = ωδε , ∂X

the variation (3.36) reduces to t2

δA = ∫ ∫ { t1 R

∂L 󵄨󵄨󵄨󵄨 󵄨 ω[X − o] + e:ω} δε dV dt . ∂X 󵄨󵄨󵄨f

The equality δA = 0 must be valid for each spatial rotation. Hence, ∂L 󵄨󵄨󵄨󵄨 󵄨 ω[X − o] + e:ω = 0 . ∂X 󵄨󵄨󵄨f By the identities

∂L 󵄨󵄨󵄨󵄨 ∂L 󵄨󵄨󵄨󵄨 󵄨󵄨 ω[X − o] = {(X − o) ⊗ 󵄨 } :ωT , ∂X 󵄨󵄨f ∂X 󵄨󵄨󵄨f

86 | 3 Essentials of Non-Linear Elasticity Theory and e:ω = e T :ωT , we have ∂L 󵄨󵄨󵄨󵄨 󵄨 + e T } :ω = 0 . ∂X 󵄨󵄨󵄨f

{(X − o) ⊗

Since ω is arbitrary antisymmetric linear mapping, we finally arrive at Asym {(X − o) ⊗

∂L 󵄨󵄨󵄨󵄨 󵄨 + eT } = 0 , ∂X 󵄨󵄨󵄨f

where Asym[. . .] is the antisymmetric part of the linear mapping [. . .]. Now consider the particular case of variation, when coordinates are fixed, while physical fields are varied, but, in contrast with Section 3.8.3, their variation is defined by a Lie group. IV. Translation of the deformation. Let ̃γ (X, t, ε) = γ(X, t) + εγ 0 , where γ0 ∈ V is a ̃ constant vector, and let X(X, t, ε) = X, ̃t(X, t, ε) = t. Since d δγ = 0 , dt

δγ = γ 0 δε ,

∂ δγ = 0 , ∂X

the variation (3.36) reduces to t2

δA = ∫ ∫ { Lx |f ⋅ γ 0 δε} dV dt . t1 R

The equality δA = 0 must be valid for each such translation. Since γ0 is arbitrary, we arrive at the following condition: Lx |f = 0 . V. Rotations of deformation. Let ̃γ(X, t, ε) = o + Q(ε)[γ(X, t) − o], where o ∈ E is a fixed point, and ℝ ∋ ε 󳨃→ Q(ε) ∈ O(3) is smooth field of orthogonal mappings such ̃ that Q(0) = I. Spatial and temporal points are fixed: X(X, t, ε) = X, ̃t(X, t, ε) = t. Like in III, we obtain δγ = ω[γ(X, t) − o]δε ,

d ∂γ δγ = ω [ ] δε , dt ∂t

∂ ∂γ δγ = ω δε , ∂X ∂X

where ω is antisymmetric linear mapping as in III. In this case, the variation (3.36) reduces to t2

δA = ∫ ∫ { Lx |f ⋅ ω[γ(X, t) − o] + p ⋅ ω [ t1 R

∂γ ∂γ ] − P: (ω )} δε dV dt . ∂t ∂X

The invariance condition is as follows: Lx |f ⋅ ω[γ(X, t) − o] + p ⋅ ω [

∂γ ∂γ ] − P: (ω )=0. ∂t ∂X

3.9 Constitutive Relations |

By the identities

87

u ⋅ ω[v] = (v ⊗ u ♭ ):ωT , L:(ωM) = (MLT ):ωT ,

which are hold for all vectors u, v ∈ V and all linear mappings L, M, ω ∈ Lin(V; V), one has ∂γ ∂γ T 󵄨 ⊗ p♭ − P } :ω = 0 . {(γ(X, t) − o) ⊗ L♭x 󵄨󵄨󵄨󵄨f + ∂t ∂X Hence, ∂γ ∂γ T 󵄨 Asym {(γ(X, t) − o) ⊗ L♭x 󵄨󵄨󵄨󵄨f + ⊗ p♭ − P }=0. ∂t ∂X

3.9 Constitutive Relations 3.9.1 Principle of Material-Frame Indifference We will consider the changes of frame (x, t) 󳨃→ φ12 (x, t) = (x∗ , t∗ ) , defined by the relations (2.31) and (2.32) x∗ = a(t) + Q(t)(x − b) , t∗ = t − c , and will assume that mappings a and Q are smooth. A change of frame defines oneparametric families of transformations on scalars, vectors, and second-rank tensors: (i) Scalars remain unchanged. (ii) Let v ∈ V. It can be represented by a point difference v = x − y in the frame φ1 . In the frame φ2 , the following relations hold: x∗ = a(t) + Q(t)(x − b) , and for v∗ = x∗ − y∗ , we have

y∗ = a(t) + Q(t)(y − b) ,

v ∗ = Q(t)v .

(3.38)

The last equality defines the one-parametric family {f t }t∈T of vector transformations f t : V → V , v 󳨃→ v ∗ . (iii) Let T ∈ Lin(V; V). If w = T[v], then using relations v∗ = Q(t)v and w ∗ = Q(t)w one arrives at the equality w ∗ = T ∗ [v ∗ ], where T ∗ = Q(t)T ∗ QT (t) .

(3.39)

We obtained the family {g t }t∈T of tensor transformations g t : Lin(V; V) → Lin(V; V) ,

T 󳨃→ T ∗ .

88 | 3 Essentials of Non-Linear Elasticity Theory

Mappings whose values are scalars, vectors, or tensors are called frame indifferent or objective [37] if both dependent and independent variables transform according to the expressions (2.31), (2.32), (3.38), and (3.39) under a change of frame. That is, let h, v and T be scalar, vector, and tensor fields relative to φ1 , and h∗ , v∗ and T ∗ be scalar, vector, and tensor fields relative to φ2 . These fields are frame indifferent, if h∗ (x∗ , t∗ ) = h(x, t) , v∗ (x∗ , t∗ ) = Q(t) v(x, t) , T ∗ (x∗ , t∗ ) = Q(t) T(x, t) Q T (t) ,

(3.40)

where (x, t) and (x∗ , t∗ ) are connected by (2.31) and (2.32). Let us consider a motion in a frame φ1 , defined by the mapping γ : SR × 𝕋 → E ,

(X, t) 󳨃→ x .

To the reference shape SR considered in φ1 there corresponds a shape SR∗ , considered in φ2 . According to (2.31) and (2.32), these shapes are related by the equalities X ∗ = a(̃t) + Q(̃t)(X − b), and ̃t∗ = ̃t − c, in which ̃t is fixed. The same motion is described in a frame φ2 by x∗ = γ∗ (X ∗ , t∗ ) = a(t) + Q(t) (γ(X, t) − b) ,

t∗ = t − c .

(3.41)

The equality (3.41) allows us to obtain the relation between deformation gradients ∂γ , and the F and F ∗ . The former one is calculated relatively to the frame φ1 , i.e., F = ∂X

latter one is calculated relative to the frame φ2 , i.e., F ∗ = follows that ∂γ ∂X F ∗ = Q(t) , ∂X ∂X ∗ and, hence, F ∗ = Q(t)F Q T (̃t) .

∂γ ∗ ∂X ∗ .

From the chain rule, it

(3.42)

Now we give a definition to the physical processes, which are the “same”. By a dynamical process, we mean an ordered pair (γ, T), consisting of the time-dependent deformation γ and the Cauchy stress tensor T. Two dynamical processes (γ, T) and (γ ∗ , T ∗ ) given in different rigid frames are called equivalent processes, if they are related to each other by a change of frame (3.40) and (3.41). Experience shows that two solids with the same reference shape and the same mass distribution may react quite differently under the influence of the same forces. This difference is described in the following manner: we say that the two solids consist of different materials. The material properties of a solid are described as conditions that give a restriction on possible dynamical processes. These conditions are called constitutive equations or constitutive assumptions. In this book, we consider constitutive equations involving only motion and the Cauchy stress tensor.

3.9 Constitutive Relations

| 89

We state the fundamental physical principle according to which material properties are indifferent, i.e., independent of the frame of reference. In mathematical terms, this means that the constitutive equations are subject to the following¹⁴ Principle of material frame indifference. Constitutive equations must be invariant under change of a frame of reference. If a constitutive equation is satisfied for a process with a motion and a Cauchy stress tensor given by x = γ(X, t) ,

T = T(X, t) ,

then it must be satisfied also for any equivalent process (γ ∗ , T ∗ ).

3.9.2 The Cauchy Polar Decomposition Theorem Further, we will use the important result obtained Cauchy: polar decomposition theorem. Theorem (Polar decomposition theorem). For any linear mapping T ∈ Inv(V; V), there exists a unique orthogonal linear mapping R ∈ O(3) and positive-definite¹⁵ symmetric linear mappings U, V ∈ InvLin(V; V), such that the following equalities hold: T = RU ,

T = VR .

Here U 2 = T T T, and V 2 = TT T . Remark 3.11. Proof of this theorem may be found in [82] and is provided as follows. The transformation T T T is positive¹⁶ and one can define its square root U := √T T T, which is invertible and symmetric. Now, take R = TU −1 . Since RRT = TU −1 U −T T T = T(U 2 )−1 T T = I , one has R ∈ O(3). Thus, we have obtained the desired decomposition T = RU. The second case is considered analogously. Cauchy–Green measures. With application to the deformation gradient F, the polar decomposition theorem asserts that there exists a unique orthogonal linear mapping

14 We give a formulation similar to Truesdell and Noll in the book [37]. 15 A linear map L is positive definite if u ⋅ L[u] > 0 for any non-zero vector u. For a positive definite operator the notion of square root is defined (Chapter 12, Section 12.5.4). 16 Indeed, for any vector u ≠ 0, and one has u ⋅ T T T[u] = T[u] ⋅ T[u] ≥ 0 . Since T is invertible, T[u] ≠ 0 and the sign ≥ can be changed by >.

90 | 3 Essentials of Non-Linear Elasticity Theory R ∈ O(3), such that¹⁷ det R = 1, and positive-definite symmetric linear mappings U, V ∈ Inv(V; V) called right and left stretch tensors, respectively, such that F = RU ,

F = VR .

(3.43)

Right and left stretch tensors define right and left Cauchy–Green tensors: C := U 2 = F T F ,

and

B := V 2 = FF T .

Let us consider the mechanical sense of the equalities (3.43). According to the equality (3.6), the deformation gradient maps non-deformed infinitesimal fiber into a deformed one. The deformation of the fiber decomposes on rotation, defined by R, and stretch, defined by U or V. Rotation is “rigid motion”, which can be seen by the observer associated with the space E. Another observer, rotating with the considering fiber, will see stretches only. Thus, objective quantities, which can be set as measures of deformation, are represented by U and V, or, equivalently, by right and left Cauchy– Green tensors C and B.

3.9.3 Simple Material Let Sh(E) be the set of all shapes in E. Define a family {gSR }SR ∈Sh(E) of mappings gSR : SR × Lin(V; V) → Sym(V) , which are called response functionals. Such a family represents the mechanical behavior of a solid. The constitutive relation of the solid with simple material relative to the reference shape SR is the following relation: T X = gSR (X, F X ) .

(3.44)

̃R be any two reference shapes and S be some actual shape. Let Let SR and S ̃R → S be deformations and λ = ̃γ−1 ∘ γ : SR → S ̃R . The following γ : SR → S and ̃γ : S diagram illustrates the relations between γ, ̃γ, λ, and their deformation gradients: γ

SR

S

−1

̃F

−1

̃γ

V ̃ F

∘F

∘γ

̃R S

17 Since det F > 0.

󳨐⇒

P=

λ=

̃γ

F

V

V

3.9 Constitutive Relations

| 91

Thus one has the following relation: gSR (X, F X ) = gSR (X, F̃ λ(X) P X ) . ̃ ̃ ), i.e., ̃ F The right-hand side of this equality may be denoted by g̃S̃R (X, X ̃ ̃ ) := gS (λ−1 (X), ̃ ̃ P −1 ̃ ) . ̃ F ̃ F g̃ ̃ (X, SR

X

R

X

λ ( X)

Thus, we have the response functional g̃S̃R . However, we need to reflect that we are dealing with the same solid. Hence, it should be g̃S̃R = gS̃R . This motivates the following. ̃R response functionals gS and g ̃ are related by Axiom. For any shapes SR and S R SR ̃ ̃ ) = gS (X, F X ) , where X ̃ = FP −1 . ̃ F ̃ = λ(X) , F g ̃ (X, (3.45) SR

X

R

Remark 3.12. We can consider SR as a “test” reference shape. Relative to this shape response functional gSR was obtained from experiments. Then, according to the relation (3.45) response functionals for the same solid relative to another reference shapes can be built. Not all mappings gSR satisfy the principle of material-frame indifference. To obtain the appropriate type of such mappings, let Q be an orthogonal tensor. Then, according to (3.40) and the principle of material-frame indifference¹⁸ we have gSR∗ (X ∗ , F ∗X ∗ ) = Q(t)gSR (X, F X )Q T (t) . The left-hand side of the equality may be considered as the value of the response functional that is relative to the reference shape SR∗ . According to the axiom, taking P = Q(̃t), we arrive at the following equality: gSR (X, F ∗X ∗ Q(̃t)) = Q(t)gSR (X, F X )QT (t) . Taking into account the equality (3.42) and the identity Q T (̃t)Q(̃t) = I, we obtain the following expression: gSR (X, Q(t)F X ) = Q(t)gSR (X, F X )Q T (t) , which holds for each orthogonal tensor Q. Now, let F = RU be a polar decomposition (see (3.43)). Taking Q = RT leads to the final expression: gSR (X, U X ) = R T gSR (X, F X )R .

(3.46)

This expression defines the functional restriction on possible mappings gSR and reflects the requirement from the principle of material-frame indifference. From (3.46), it follows that T X = RgSR (X, U X )RT . (3.47) The reference shape SR is stress free, if gSR (X, I X ) = 0 for each X ∈ SR . The assumption of the existence of a stress-free reference shape for any solid is traditional in conventional elasticity.

18 We consider equivalent processes. According to the principle of material-frame indifference since T X = gSR (X, F X ), then T ∗X = gS∗ (X ∗ , F ∗X ), and symbol g is the same. R

92 | 3 Essentials of Non-Linear Elasticity Theory

3.9.4 Representation Theorems Suppose that V is an n-dimensional Euclidean vector space. We will consider mappings ε : Lin(V; V)k → ℝ and f : Lin(V; V)k → Lin(V; V), where Lin(V; V)k = Lin(V; V) × ⋅ ⋅ ⋅ × Lin(V; V) (k times). Our goal is to introduce the notion of isotropic mapping [37]. Mapping ε : (A 1 , . . . , A k ) 󳨃→ ε(A 1 , . . . , A k ) is called isotropic, if the following equality ε(A 1 , . . . , A k ) = ε (QA 1 QT , . . . , QA k QT ) holds for all orthogonal mappings Q and all tuples (A 1 , . . . , A k ) from the domain of ε. Isotropic mapping ε : A 󳨃→ ε(A) of one variable is called the orthogonal invariant (or, simply, invariant) of A. In the case of several variables, ε is called a simultaneous invariant. The following theorem sets necessary and sufficient conditions for ε to be invariant: Theorem. A mapping ε : Sym(V) → ℝ is isotropic (invariant) iff it can be represented as a function of the principal invariants of its argument. That is, ε(A) = ̃ε (I1 (A), I2 (A), I n (A)) . Mapping f : (A 1 , . . . , A k ) 󳨃→ f (A 1 , . . . , A k ) is called isotropic, if the following equality: Qf (A 1 , . . . , A k )Q T = f (QA1 QT , . . . , QA k QT ) holds for all orthogonal mappings Q and all tuples (A 1 , . . . , A k ) from the domain of f . Now, we formulate the classical representation theorem. It is formulated for the general case of n-dimensional real vector space V. Theorem (The Rivlin and Ericksen representation theorem). A mapping f : A 󳨃→ f (A), defined on symmetric tensors, is isotropic iff it can be expressed as follows: D = f (A) = φ0 I + φ1 A + φ2 A 2 + ⋅ ⋅ ⋅ + φ n−1 A n−1 . φ k (I1 (A), . . . , I n (A)). Here φ k = ̃ Proof¹⁹. Let A be a symmetric tensor. Consider an eigenvector e of the tensor A. If U = {xe | x ∈ ℝ} denotes the one-dimensional vector space generated by e, then one has the direct sum V = U ⊗ U⊥ of the line U and the plane U⊥ . Define orthogonal transformation Q for which U⊥ is a fixed set²⁰: {−u, Qu = { u, {

u∈U, u ∈ U⊥ .

19 We follow [37]. 20 Q is a reflection on the plane U⊥ . If one choose a basis (e i )ni=1 for V, where e1 = e, and (e i )ni=2 is some basis for U⊥ , then [Q i j ] = diag {−1, 1, . . . , 1}.

3.9 Constitutive Relations |

93

According to the definition²¹, QAQT = A. Since f is an isotropic function, QDQT = D or QD = DQ. Thus, Q(De) = D(Qe) = −De . Hence, De ∈ U. That is, De = de for some d ∈ ℝ. This means that e is an eigenvector for D. Another eigenvectors of A are considered in an analogous way. Thus, one has that all eigenvectors of A are eigenvectors of D. Denote all distinct eigenvalues of A by a1 , . . . , a m , where m ≤ n, and the corresponding eigenvectors by e1 , . . . , e m . These vectors are similarly eigenvectors for D, which correspond to eigenvalues d1 , . . . , d m . Note that these numbers may be not distinct. The system of m equations d k = φ0 + φ1 a k + φ2 a2k + ⋅ ⋅ ⋅ + φ m−1 a m−1 , k

k = 1, . . . , m ,

has the unique solution {φ0 , φ1 , . . . , φ m−1 }, since its determinant is equal to ∏(a j − a k ) ≠ 0 . j 0, and a one-parametric family {γ ε }ε∈]−a,a[ , γ ε : SR → Sε of deformations. We suppose that this family satisfies the following properties: (i) S0 = SR and γ0 = IdSR : SR → SR is the identity map; (ii) for each X ∈ SR the mapping ε 󳨃→ γ ε (X) from ] − a, a[ to E is smooth.

104 | 3 Essentials of Non-Linear Elasticity Theory First-order Taylor expansion of ε 󳨃→ γ ε (X) in a neighborhood of ε = 0 has the form: γ ε (X) = γ0 (X) + ε

∂γ ε (X) 󵄨󵄨󵄨󵄨 + O(ε2 ) as 󵄨 ∂ε 󵄨󵄨󵄨ε=0

ε→0.

Since γ0 (X) = X, one can denote γ ε (X) − X by u ε (X). This vector represents a displacement from the reference place X to the actual one, γ ε (X), which corresponds to the “infinitesimally” distorted shape Sε . One has the formula u ε (X) = εs(X) + O(ε2 )

as

ε→0,

∂γ ε (X) 󵄨󵄨󵄨󵄨 󵄨 represents a rate of “perturbation” at point X. Let ∂ε 󵄨󵄨󵄨ε=0 ̃ ε := εs. It is the principal linear part of u ε , an infinitesimal displacement. us denote u The displacement gradient of u ε at point X ∈ SR has the form: where the vector s(X) :=

u 󸀠ε (X) = H ε (X) + O(ε2 ) as

ε→0.

̃ ε: Here, H ε (X) = εs 󸀠 (X) is the gradient of u ̃ ε (X + h) = u ̃ ε (X) + H ε (X)[h] + o(‖h‖) u

as

h→0,

We introduce the infinitesimal strain and rotation measures (vorticity tensor): ̃ ε = 1 (H ε + H T ) , E ε 2

1 ̃ R ε = (H ε − H Tε ) . 2

̃ε + ̃ ̃ 󸀠ε = E Since u R ε , it follows that ̃ε + ̃ u 󸀠ε = E R ε + O(ε2 )

as

ε→0.

With accuracy up to O(ε2 ), the exact polar decomposition F ε = R ε U ε can be approxĩε + ̃ ̃ 󸀠ε = E mately replaced by the additive decomposition u Rε . We obtain the asymptotic expansions for the right Cauchy–Green strain C ε , the right and left stretch tensors U ε and V ε , and the orthogonal tensor R ε . According to (3.17), one has the asymptotic expansions C ε = I + (H ε + H Tε ) + O(ε2 ) as B ε = I + (H ε +

H Tε )

2

+ O(ε ) as

ε→0, ε→0,

for the right and left Cauchy–Green strain measures. From the theorem of polar decomposition, it follows that F ε = I + u 󸀠ε = R ε U ε , where U ε = √C ε = I +

1 (H ε + H Tε ) + O(ε2 ) 2

as

ε→0.

3.11 Linearized Elasticity

| 105

Thus, one has the following expansion for the orthogonal tensor R ε : Rε = I +

1 (H ε − H Tε ) + O(ε2 ) 2

as

ε→0.

̃ ε and ̃ In terms of E R ε , the previous formulae look as follows: ̃ ε + O(ε2 ) , Uε = I + E

Rε = I + ̃ R ε + O(ε2 )

as

ε→0.

(3.54)

Finally, from the theorem of polar decomposition, one also has F ε = V ε R ε , and then ̃ ε + O(ε2 ) as Vε = I + E

ε→0.

3.11.2 Linearized Constitutive Relations Let the material be elastic, i.e.. Its constitutive relation has the form T ε = R ε gSR (U ε )R Tε , where mapping gSR : SimLin(V; V) → SimLin(V; V) depends on the shape SR ; F ε = R ε U ε . We will obtain asymptotic relations by virtue of (3.54). Using the first-order Taylor formula, one obtains ̃ ε + O(ε2 )) = gS (I) + LS [E ̃ ε ] + O(ε2 ) as gSR (U ε ) = gSR (I + E R R where LSR =

ε→0,

∂gSR (U) 󵄨󵄨󵄨󵄨 : SimLin(V; V) → SimLin(V; V) . 󵄨 ∂U 󵄨󵄨󵄨U=I

Denoting T 0 := gSR (I) and performing calculations ̃ ε ] + O(ε2 ))(I − ̃ T ε = R ε gSR (U ε )R Tε = (I + ̃ R ε + O(ε2 ))(T 0 + LSR [E R ε + O(ε2 )) , one arrives at the formula ̃ ε ] + O(ε2 ) Rε T0 − T0 ̃ T ε = T0 + ̃ R ε + LSR [E

as

ε→0.

(3.55)

Here, T 0 is the Cauchy stress tensor in the shape SR . The fourth-rank tensor LSR is called the elasticity tensor. If SR is stress free, then T 0 = 0, and ̃ ε ] + O(ε2 ) T ε = L[E

as

ε→0,

which represents the classical Hooke’s law with accuracy up to O(ε2 ). We suppressed the lower index of L in this case, since the latter relation is special: it is valid only for stress-free reference shapes.

106 | 3 Essentials of Non-Linear Elasticity Theory

3.11.3 Linearized Stresses Let us denote

̃ ε] , ̃ Rε T0 − T0 ̃ R ε + LSR [E Tε = T0 + ̃

then according to (3.55), one has Tε = ̃ T ε + O(ε2 )

ε→0.

as

T ε may be considered The latter relation shows that in the case of T 0 = 0, the tensor ̃ as the principal linear part of the Cauchy stress tensor T ε . Finally, one obtains that in the linear elastic case, respectively to a stress free ref̃ ε and ̃ erence shape, fields u T ε play the role of “displacement” and “stress” fields, respectively. One can linearize the equation of motion and boundary conditions and express them in terms of the so-obtained fields. Remark 3.14. After linearization, the equation of motion, div T ε + b = ρ ü ε , takes the form 1 T ̃ 󸀠ε + [u ̃ 󸀠ε ] ] + b = ρ u ̃̈ ε , div L [u 2 with accuracy up to O(ε2 ). In the isotropic case, one obtains the Lamé equation. Now, we consider asymptotic relations for Piola–Kirchhoff stress tensors in the linear elastic case. First, we obtain the asymptotic expansion of J ε = det F ε . Since F ε = I +u 󸀠ε , one has d det A 󵄨󵄨󵄨󵄨 󵄩 󵄩2 J ε = det(I + u 󸀠ε ) = det I + [u 󸀠 ] + o (󵄩󵄩󵄩󵄩u 󸀠ε 󵄩󵄩󵄩󵄩 ) , 󵄨 dA 󵄨󵄨󵄨A=I ε A 󵄨󵄨󵄨 where d det dA 󵄨󵄨 A=I ∈ Lin(Lin(V; V); ℝ). In terms of ε, d det A 󵄨󵄨󵄨󵄨 ̃ 󸀠 ] + O(ε2 ) as ε → 0 . J ε = det I + [u 󵄨 dA 󵄨󵄨󵄨A=I ε Remark 3.15. Further, we use the following classical result. Let U be an n-dimensional vector space, A, B ∈ Lin(U; U). Then, d det A [B] = det A tr{A−1 B} . dA Finally, one arrives at the relation ̃ 󸀠ε + O(ε2 ) J ε = 1 + tr u

as

ε→0,

̃ 󸀠ε , or, according to the additive decomposition of u ̃ ε + O(ε2 ) J ε = 1 + tr E

as

ε→0.

3.11 Linearized Elasticity

|

107

Since P ε = J ε T ε F −T ε , and taking into account that J ε = 1 + O(ε) ,

F −T ε = I + O(ε) ,

Tε = ̃ T ε + O(ε2 ) as

ε→0,

̃ ε ], one obtains the following relation for the first Piola–Kirchhoff stress and ̃ T ε = L[E tensor: Pε = ̃ T ε + O(ε2 ) as ε → 0 . The relations obtained for the Cauchy and Piola–Kirchhoff stress tensors show that with accuracy up to O(ε2 ), respectively to a stress-free reference shape, these fields are equal to ̃ T ε.

3.11.4 The Green–Rivlin–Shield–Truesdell Formula Suppose that SR is a stress-free shape and γ : SR → S is a deformation onto some shape S; its deformation gradient is F. Consider “infinitesimal” perturbations of the latter shape. Analogously to 3.11.1, we introduce a one-parametric family {Sε }ε∈]−a,a[ of shapes, a > 0, and the corresponding one-parametric family {γ ε }ε∈]−a,a[ ,

γ ε : S → Sε

of deformations, such that: 1) S0 = S and γ0 = IdS : S → S; 2) for each x ∈ S the mapping ε 󳨃→ γ ε (x) from ] − a, a[ to E is smooth. Relations between γ, γ ε and the corresponding shapes are shown in the diagram: γ

SR

S

γε

γε

∘γ Sε First-order Taylor expansion of ε 󳨃→ γ ε (x) in a neighborhood of ε = 0 has the form: γ ε (x) = γ0 (x) + ε

∂γ ε (x) 󵄨󵄨󵄨󵄨 + O(ε2 ) as 󵄨 ∂ε 󵄨󵄨󵄨ε=0

ε →0.

The difference between points x, γ ε (x) ∈ E defines the translation vector u ε (x) := γ ε (x) − x, the displacement of the particle that occupied place x in the shape S, and the place γ ε (x) in the shape Sε . The differential of the mapping ε 󳨃→ u ε (x) is of the form ∂γ ε (x) 󵄨󵄨󵄨󵄨 ̃ ε (x) := εs(x) , where s(x) := , u 󵄨 ∂ε 󵄨󵄨󵄨ε=0 and represents the linear part of the displacement. The vector s(x) is the rate of change of this linear part with respect to ε.

108 | 3 Essentials of Non-Linear Elasticity Theory ̃ ε (x) is the field of infinitesimal displacements; For fixed ε, the mapping S ∋ x 󳨃→ u denote its gradient by H ε : x 󳨃→ H ε (x), i.e., ̃ ε (x + h) = u ̃ ε (x) + H ε (x)[h] + o(‖h‖) u

as

h→0,

̃ ε and ̃ Rε : and the corresponding infinitesimal strains and vortices by E ̃ ε = 1 (H ε + H T ) , E ε 2

1 ̃ R ε = (H ε − H Tε ) . 2

The total deformation is the composition χ ε := γ ε ∘ γ, and its gradient is the linear map F ε = H ε F. The corresponding left Cauchy–Green strain tensor, B ε = F ε F Tε and the related quantities can be represented for small ε by the following asymptotic expansions: B ε ≐ B + H ε B + BH Tε , B = FF T , −1 − B −1 H ε − H Tε B−1 . B−1 ε ≐ B

Here, the symbol ≐ indicates equality with up to O(ε2 ). The material is assumed to be isotropic, obeying the relation (3.48): T = ℶ0 I + ℶ1 B + ℶ−1 B−1 , where ℶr , r = −1, 0, 1, are functions on the principal invariants I k , k = 1, 2, 3, of B. This constitutive relation is written relatively to the stress-free shape SR . In the case of small ε, the following asymptotic expansion holds: 3

ℶr,ε ≐ ℶr + ∑ k=1

∂ℶr dI k ( {H ε B + BH Tε }) . ∂I k dB

Using formulae for the principal invariants derivatives are [37] dI1 (B) dI2 (B) [A] = tr(A) , [A] = tr [(I1 (B)I − BT ) AT ] , dB dB dI3 (B) [A] = tr [(I3 (B)(B −1 )T ) AT ] , dB and substituting the relations obtained in (3.48), we obtain T ε ≐ T 0 + ℶ1 (H ε B + BH Tε ) − ℶ−1 (B−1 H ε + H Tε B−1 ) 1

+ ∑ ( r=−1

+

∂ℶr ∂ℶr tr (H ε B + BH Tε ) + tr (I1 (B)H ε B + I1 (B)BH Tε − BH ε B − B2 H Tε ) ∂I1 ∂I2

∂ℶr tr (I3 (B)B −1 H ε B + I3 (B)H Tε )) B r , ∂I3 (3.56)

where T 0 = ℶ0 I + ℶ1 B + ℶ−1 B−1 .

3.12 Distributed Defects in Solids

|

109

The expression (3.56) can be simplified if one uses the identity ℶ1 (H ε B + BH Tε ) − ℶ−1 (B−1 H ε + H Tε B−1 ) ̃ ε B−1 + B−1 E ̃ ε + ℶ−1 {E ̃ ε }) , = H ε T 0 + T 0 H Tε − 2 (ℶ0 E and the Cayley–Hamilton theorem, from which follows B2 = I1 B − I2 I + I3 B−1 . As a final result, we obtain the Green–Rivlin–Shield–Truesdell formula [89]: ̃ ε] , T ε ≐ T0 + ̃ Rε T0 − T0 ̃ R ε + L γ [E

(3.57)

where L γ is the elasticity tensor ̃ ε] = E ̃ ε T0 + T0 E ̃ ε − 2 (ℶ0 E ̃ ε + ℶ−1 {E ̃ ε }) ̃ ε B−1 + B−1 E L γ [E 1

+ 2 ∑ {(I2 r=−1

∂ℶr ∂ℶr ̃ ε + ∂ℶr tr(B E ̃ ε ) − I3 ∂ℶr tr (B−1 E ̃ ε )} B r . + I3 ) trE ∂I2 ∂I3 ∂I1 ∂I2

3.12 Distributed Defects in Solids 3.12.1 Preliminary Remarks The violation of the compatibility conditions characterizes the phenomenon of plastic deformation, which is a result of the development and redistribution of defects in the ideal body (crystal). A physical interpretation of the plastic deformation of crystals was formulated by Dehlinger, Taylor, Orowan, and Polanyi [90–94], who explained the divergence between the theoretical assessment of mechanical properties of crystals, obtained by Frenkel [95], and the experimental data. It should be mentioned that within the framework of these ideas, the plastic distortion βp was considered to be known, whereas its symmetric part εp was believed to be incompatible. The measure of incompatibility was defined by the incompatibility tensor η = −∇ × (∇ × εp )T , or the dislocation density tensor α = −∇ × βp [66]. The latter characteristic has a clear geometrical interpretation as a subintegral expression for determining the Burgers vector [96]. In this case, the stressed state in the body is fully determined by elastic deformation, imposed on the plastic deformation such that the body does not split into parts [96]. This allows classification of the body as a local simple body. It should → 󳨀 be noted that elastic deformation and rotation are interrelated as (∇ × ε)T + ∇ ω = K, where K is the Cosserat tensor characterizing the relative rotation of the crystalline lattice. Thus, due to incompatible plastic deformation, the body is divided into a puzzle of infinitesimally small elements (fragments) that can be combined (assembled) as a solid body by means of elastic individual deformation of each such fragment. A more general situation occurs if we believe that, besides individual deformation,

110 | 3 Essentials of Non-Linear Elasticity Theory

each fragment should be imparted an individual rotation not connected with deformation, thereby, defining the orientation of the directors (in the terminology of the theory of moment media). To this end, the bending-torsion tensor κ should be used, as proposed by De Wit; in the case of compatible deformation, this tensor is defined → 󳨀 by the equality κ = ∇ω, and in the general case, it is specified as an arbitrary smooth field [66]. In such a case, the stressed state of the body depends on two independent fields, i.e., the elastic deformation and the elastic bending-torsion, which ensure integrability of the relations for full fields. The sources of internal stresses are determined by deformation incompatibility measures (the density of the dislocations α) and the measure of disclination density, determined by the equality θ = −∇ × κp . Speaking figuratively, owing to the continuous distribution of the two types of defects, i.e., dislocations and disclinations, the body is divided into a puzzle of fragments that can be assembled as a solid body by means of elastic individual deformation and independent elastic orientation of the edges. In the framework of this approach, elastic distortion turns out to be indefinite, although its symmetric part is determined definitively. The stressed state of the body is characterized by the laws of state for media with moment stresses [97] or a microstructure [98]. Experimental identification of disclinations was done in the mid-twentieth century [99–101], in particular, in investigations of liquid crystal mechanics. It should be noted that pioneering works [66, 96] pointed to a close link between dislocations and disclination density tensors and the vectors of curvature and torsion of material connection. It also pointed to a link between the continuum theory of dislocations and the Einstein–Cartan theory of gravitation.

3.12.2 Total Distortion 1-Forms The response of a conventional solid is determined with respect to a reference shape that is supposed to be embedded into three-dimensional Euclidean space in a relaxed state, i.e., such a reference shape has to be free of stresses. In that case, the localization of deformation γ is defined by exact 1-forms F i = dγi , i = 1, 2, 3. The starting point in the construction of the theory of continuously distributed defects is the rejection of this proposal. The exact forms F i ∈ Ω1 (SR ) are replaced by some, not necessarily exact, forms H i ∈ Ω1 (SR ), i.e., in general, dH i ≠ 0. The only condition that should be demanded is that H 1 ∧H 2 ∧H 3 ≠ 0. We will refer to the 1-forms H i = H ia dx a as 1-forms of total distortion and consider them as being both spatial and time dependent. Remark 3.16. We use fraktur notation i for the index in F i and H i on purpose to designate that F i and H i are not components of a vector. Both F i and H i are 1-forms.

3.12 Distributed Defects in Solids

|

111

3.12.3 Four-Dimensional Space and the Homotopy Operator Frames. LetE 3 be a three-dimensional Euclidean affine space. Here, and until the end of this section, we use capital Latin letters to designate indices that vary from 1 to 3. The volume element of E 3 is denoted by μ = dx1 ∧ dx2 ∧ dx3 . That is, μ is a frame for Ω3 (E 3 ). A coordinate frame (∂ A )3A=1 on E 3 defines the collection (μ A )3A=1 of the 2-forms μ A := ∂ A ⌟ μ, which serve as a frame for Ω2 (E 3 ). Differential forms μ A can be viewed as oriented surface elements. We will use four-dimensional Euclidean affine space E 4 = E 3 ×ℝ. Lowercase Latin letters are used as indices that vary from 1 to 4. The coordinate frame (∂ A )3A=1 on E 3 induces the coordinate frame (∂ i ) = ((∂ I )3I=1 , ∂ t ) on E 4 . Here, t is a natural coordinate on ℝ. In the following, we often will use the notation ∂4 for ∂ t . The volume element on E 4 is denoted by π and can be represented as π = μ ∧ dt . The differential form π is a frame for Ω4 (E 4 ). The frame (∂ i )4i=1 generates the differential forms π a = ∂ a ⌟ π, a = 1, 2, 3, 4. Thus, the family (π a )4a=1 forms a frame for Ω3 (E 4 ). We use two designations for the exterior derivative. The first one, d, is defined as d = dx I ∧ ∂ I , while the second one, d, is defined via d = dx i ∧ ∂ i . Operators d and d are related by d = d + dt ∧ ∂4 . Note that the operator d can be considered as a differentiation with fixed t. Remark 3.17. Note that μ 1 = dx2 ∧dx3 , μ2 = dx3 ∧dx1 , μ3 = dx1 ∧dx2 are 2-forms that can be defined equivalently as μ I = ∗(dx I ), where ∗ is the Hodge star operator²². The index I was chosen as lower by virtue of the definition μ I = ∂ I ⌟ μ. Such a designation was accepted in [102]. An analogous situation holds for π a . Star-shaped regions. Let SR ⊂ E 3 be a shape. Assume that there exists a point x0 ∈ SR , such that the line segment connecting x0 and any other point x ∈ ∂SR intersects ∂SR only at x. In this case, the set SR is called star shaped. In further reasoning, we consider only the star-shaped set SR . Homotopy operators. In order to decompose total distortion on exact and non-exact parts, we introduce a special linear map. For each p = 1, 2, 3, 4, we define the linear map H : Ω p (SR × ℝ) → Ω p−1 (SR × ℝ) such that for any p-form ω ∈ Ω p (SR × ℝ), one has [102] 1

̃ (λ) dλ , Hω = ∫ λ p−1 X ⌟ ω 0

22 See Chapter 13, Section 13.4.2.

112 | 3 Essentials of Non-Linear Elasticity Theory where X = (x i − x0i )∂ i , sum over i = 1, 2, 3, 4, and ̃ (λ) = ω i1 ...i p (x0b + λ (x b − x0b )) dx i1 ∧ ⋅ ⋅ ⋅ ∧ dx i p . ω

3.12.4 Decomposition of Total Distortion into Exact and Antiexact Parts Let H be a homotopy operator. It has the property [102] dH + Hd = IdΩ p (SR ×ℝ) . Thus, total distortion H i can be written as H i = dHH i + HdH i .

(3.58)

By the construction, dHH i is an exact part of H i that can be represented as exterior differential of some smooth function γi , i.e., dHH i = dγi . Thus, γi = HH i + hi , where dhi = 0. The second item in (3.58) gives antiexact part of H i that characterizes the distribution of defects. If we denote the space of all exact 1-forms by E1 (SR × ℝ) and the space of all antiexact 1-forms by A1 (SR × ℝ), we have the following direct sum [102]: Ω1 (SR × ℝ) = E1 (SR × ℝ) ⊕ A1 (SR × ℝ) , and the following result: if ν ∈ A1 (SR × ℝ) then ν = Hdν. In other words, Hd = IdA 1(SR ×ℝ) , and the homotopy operator H is inverse to d.

3.12.5 Continuity Equations of Defect Dynamics We introduce the following differential forms, describing the density of defects and their currents [103]: ∙ 2-Forms of dislocation density: α i = α Ai μ A . ∙

1-Form of dislocation current: J i = J iA dx A .

∙

2-Form of disclination current: Si = S Ai μ A .

∙

3-Form of disclination density: Qi = qi μ .

3.12 Distributed Defects in Solids

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113

In general, these differential forms depend on time as well as on spatial variables. Mappings α i , J i , Si and Qi satisfy the following equations [103]: ∂ t α i = −dJ i − Si ,

∂ t Qi = −dSi ,

dα i = Qi ,

dQi = 0 .

(3.59)

In a chart, these equations can be rewritten as (here e ABC is the alternator (2.4)) ∂ t α Ai = −e ABC ∂ B J ic − S Ai ,

∂ A α Ai = qi ,

∂ t qi = −∂ A S Ai ,

while the last equation in (3.59), dQi = 0, is identically satisfied, since Qi is a 3-form. The first integrals of the system (3.59) define the kinematic field equations [103]: ∂ t k i = dwi − Si ,

dk i = Qi ,

∂ t β i = dV i − J i − wi ,

dβ i = α i − k i ,

where ∙ k i = k Ai μ A is a bend-twist 2-form; ∙ wi = wiA dx A is a spin 1-form; ∙ β i = β iA dx A is a distortion 1-form; ∙ V i is a velocity zero-form. In a chart, these equations take the form ∂ t k A = −S Ai + e ABC ∂ B wic ,

∂ A k Ai = qi , e ABC ∂ B β iC = α Ai − k Ai .

∂ t β iA = ∂ A V i − J iA − wiA ,

3.12.6 4D Representation Following [102, 103], we define disclination 3-forms Ωi and dislocation 2-form Di in 4D-Euclidean space as Ωi = −Si ∧ dt + Qi = S Ai μ A ∧ dt + qi μ , Di = J i ∧ dt + α i = J iA dx A ∧ dt + α Ai μ A

(3.60)

Then, the equations (3.59) can be rewritten in a more concise form: dΩi = 0 ,

dDi = Ωi ,

(3.61)

and their first integrals can be represented as Di = dH i + K i ,

H i = V i dt + β i ,

K i = −wi ∧ dt + k i .

3.12.7 The Momentum Equation Let T be a kinetic energy density and Ψ be a potential energy density. We define 3forms in E 4 : ∂(T − Ψ) ∂T ∂Ψ πa = π4 − πA . zi = ∂H ia ∂H i4 ∂H iA

114 | 3 Essentials of Non-Linear Elasticity Theory

Following conventional terminology, we will refer to pi :=

∂T

,

∂H i4

as components of momentum, and to σ iA :=

∂Ψ ∂H iA

,

as components of Piola–Kirchhoff stresses. Then, zi = −σ iA π A + pi π4 = −σ iA μ A ∧ dt + pi μ . The balance of momentum is tantamount to affirming that zi are exact 3-forms, i.e., dzi = (∂4 pi − ∂ A σ iA ) π = 0 .

(3.62)

3.12.8 Equations in Matrix Form Equations (3.59) and (3.62) can be rewritten in elegant matrix form. If we define the matrices H1 H = (H 2 ) , H3

k1 K = (k2 ) , k3 Z = (z1

D1 D = (D2 ) , D3

Ω1 Ω = (Ω2 ) , Ω3

z3 ) ,

z2

then equations (3.59) and (3.62) take the form dΩ = 0 ,

Ω = dD = dK ,

D = dH + K ,

dZ = 0 .

3.12.9 Relations of Dislocation and Disclination Forms with Connection, Curvature, and Torsion Forms Dislocation 2-forms D i and disclination 3-forms Ω i are closely related with the connection 1-form Γ and the corresponding curvature 2-form Θ, torsion 2-form Σ, and soldering 1-form ν. In order to show this, recall the Cartan structural equations dν = −Γ ∧ ν + Σ , dΣ = −Γ ∧ Σ + Θ ∧ ν , dΓ = −Γ ∧ Γ + Θ , dΘ = Θ ∧ Γ − Γ ∧ Θ .

(3.63)

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These equations can be formally resolved with respect to ν, Σ, Γ, and Θ by applying the homotopy operator H: ν = A[dγ + η − H(Γ A ∧ dγ)] , Σ = A[dη + Γ A ∧ η + H(dΓ A ∧ dγ) − Γ A ∧ H(Γ A ∧ dγ)] , Γ = AΓ A A−1 − dAA−1 , Θ = A(dΓ A + Γ A ∧ Γ A )A−1 , where A is a solution of the matrix integral Riemann–Graves equation A = I − H(Γ A ) while γ, η and Γ A are determined from the relations γ = γ0 + H(A−1 ν) , η = H(A−1 Σ) , Γ A = H(A−1 ΘA) . Let us identify total distortion 1-form H i with soldering 1-form ν and dislocation 3-form with torsion 2-form Σ. Then, from (3.63) and (3.60), it follows that Ω=Θ∧ν−Γ∧Σ,

(3.64)

K =Γ∧ν.

Applying the homotopy operator H to the first two equations (3.63), we obtain explicit representations for H i and D i : H = A(dγ + η − H(Γ A ∧ dγ)) , D = A(dη + Γ A ∧ η + H(dΓ A ∧ dγ) − Γ A ∧ H(Γ A ∧ dγ)) . Consequently, substituting these relations into (3.64), we obtain K = A (Γ A − A−1 dA) ∧ (dγ + η − H(Γ A ∧ dγ)) , Ω = A[(dΓ A − Γ A ∧ Γ A ) ∧ (dγ + η − H(Γ A ∧ dγ)) − (Γ A − A−1 dA) ∧ (dη + Γ A ∧ η + H(Γ A ∧ dγ) − Γ A ∧ H(Γ A ∧ dγ))] . Thus, all quantities are expressed in terms of η, Γ A , and A. If disclinations are neglected, then the expressions obtained above can be sufficiently simplified. In this case, Ω = 0, K = 0 and A = I. Then, we have H = dγ + η ,

D = dη ,

or

η = H(D) .

If, additionally, dislocations are also neglected, i.e., there are no defects at all, then we arrive at the conventional kinematic relations: D=K=0,

Ω=0,

ΓA = 0 ,

η=0,

A=I,

H = dη .

116 | 3 Essentials of Non-Linear Elasticity Theory

Note that the relations between differential forms, which describe the distribution of defects, and purely geometric objects – differential forms, related with connection on the principal bundle, yield from very abstract and general relations, which determine the properties of differentially closed systems, to which the Cartan structural equations belong.

3.12.10 Application of Yang–Mills Coupling Theory Consider a Lagrangian L0 (dγ) that describes the stress-strain state of a simple solid in conventional elasticity. Let A ∈ G0 be an element of the Lie group G0 that acts on γ from the left²³, i.e., γ󸀠 = A ⊳ γ. If A performs a homogeneous action, i.e., it does not depend on the spatial variables and time, then d and A commute: dγ󸀠 = d(A ⊳ γ) = A ⊳ dγ . In this case, the Lagrangian (constructed with account of frame-indifference principle) is invariant with respect to A: L0 (dγ󸀠 ) = L0 (A ⊳ dγ) = L0 (dγ) . If A performs a non-homogeneous action, then the relation between γ󸀠 and γ becomes more complex: dγ󸀠 = (dA) ⊳ γ + A ⊳ dγ , and the Lagrangian is no longer invariant under the action of A: L0 (dγ󸀠 ) = L0 ((dA) ⊳ γ + A ⊳ dγ) ≠ L0 (dγ) . The concept of minimal replacement, which is adopted in Yang–Mills theory, restores the invariance of the Lagrangian by replacing d 󳨃→ D :

dγ 󳨃→ Dγ := dγ + B[γ] .

The fields B[γ] are referred to as calibration fields. The suitable choice of calibration fields allows preserving the invariance of the Lagrangian under simultaneous replacing of γ 󳨃→ A ⊳ γ and d 󳨃→ D. In [102], it is shown that these calibration fields can be determined by some connection field Γ and torsion φ, i.e., B[γ] := Γγ + φ . This means that for given dislocation and disclination forms, one can uniquely determine the Cartan structural equations and, consequently, the connection form and

23 See Chapter 14, Section 14.1.

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the corresponding torsion, curvature, and solder form, and, after this, determine the calibration fields that compensate the alternation of the Lagrangian under nonhomogeneous action A. This action locally defines the transformation of elementary volume into the unstressed state, which is impossible for solids with distributed defects in Euclidean space. However, together with the action A, the replacement d 󳨃→ D is performed, which means the passage to non-Euclidean geometry (internal geometry) of the solid which does the transformations of elementary volumes into unstressed state is possible “in large”, i.e., for the whole solid. The concept of minimal replacement involves the definition of the extended Lagrangian according to the concept of minimal coupling [102]. The extended Lagrangian for defected simple solids can be written as L = L0 (Dγ) + s1 L1 (Γ) + s2 L2 (φ) , where L0 (Dγ) describes the action of the elastic properties of the material, L1 (Γ) describes the action of disclinations, and L2 (φ) describes the action of dislocations. Stationarity conditions of the action with respect to variation of γ give the equation DZ = −2R ∧ Θ , (3.65) under variation of Γ,

1 J, 2

(3.66)

Z = 2DR ,

(3.67)

DG = and with respect to variation of φ,

where R = ∂L ∂D (for more details, see [102]). The equations (3.65), (3.66), and (3.67) are referred to as evolution equations.

4 Geometric Formalization of the Body and Its Representation in Physical Space 4.1 Geometric Motivation In the conventional theory of elasticity, the reference shape is assumed to be free from stresses, whereas the configuration result in such a shape is called natural. The notion of natural configuration is often overlooked, yet it is intimately connected with the thermodynamical concept of the representative elementary volume (REV). In order to define macroscopic fields such as traction and stresses, one has to imagine shape of the body being subdivided on sufficiently small REVs. It is assumed that these REVs are sufficiently small in comparison with the whole shape, so that it can be considered an infinitesimal neighborhood of the position material point and sufficiently large in comparison with the material of microstructure, so that the axiom on the local thermodynamic equilibrium is valid. Then, one has to apply some fundamental principles to the arbitrary collection of REVs, such as the cut principle of Euler and Cauchy [52, p. 154], which, together with the freezing principle and Eulerian laws of dynamics, yields the definition of the macroscopic stress tensor field. Obviously, this field can be defined only up to some gauge, and therefore, the notion of “zero stresses” is also relative to this gauge. In this regard, it is seems reasonable to replace the notion of stress free REV by the notion of uniform REV, which means that the physical state of this REV, in particular, its stress state, is undistinguishable from some the standard state. Assume now that we have found the required compromise, and the body can be presented as a collection of REVs. Global deformation of the body causes local deformation of REVs. Choosing the standard body of a simple geometric shape, for example, a cube, and implementing in it homogeneous deformation, one can determine response of a standard REV on the deformation and fit the law of response, which is also referred to as the constitutive law. It is emphasized that constitutive laws are local, i.e., each of them relates the distortion of the shapes of the REV with the energy density or stresses added to it or reduced in it. Note that it is about increments of energy density (stresses), rather than about its absolute values. Now, the constitutive law fittingly obtained for a standard REV is applied to all REVs constituting the shapes of the body. For inner boundary-value problem formulation we should assume that the body “is assembled” from the same elementary volumes as the standard body, and that all these volumes are in the same physical state as the volumes of the standard body before the experiment. Then, solving the relevant boundary value problem, we can determine the distribution of the energy (stresses) density in the modeled body. Therefore, we conclude that the classical approach assumes the hypothesis: REVs constituting the standard body prior to experiment and REVs constituting the modeled body, are in the same physical state.

https://doi.org/10.1515/9783110563214-004

120 | 4 Geometric Formalization of the Body and Its Representation in Physical Space

According to the terminology of [104], the shape of the standard body and the reference shape of modeled body are uniform (or, equivalently, the shapes of these bodies are the images of the uniform configurations). Therefore, we shall hereinafter use the term “uniform reference configuration” instead of the term “natural reference configuration”. However, a uniform reference configuration does not exist in every case! The body may have a variety of reasons abolishing its uniformity, like defects, inhomogeneous temperature fields, inhomogeneous shrinkage caused by a chemical reaction, etc. Experimentally the amount of inhomogeneity can be estimated by dissection of the body’s shape into a number of small parts, allowing them to deform independently. Many experimental techniques, like the drilling method, the milling method, and others were developed around this idea [105, 106]. Theoretically, such a technique is equivalent to elimination of couplings between parts such that they can be free and become uniform; however, taken as a whole, they are no longer a solid body. One can imagine these “relaxed” parts as a mosaic that cannot be folded without gaps and overlaps in the Euclidean space. The modeling of such discontinuous shapes within the framework of conventional continuum mechanics represents a substantial problem. There are, at least, two ways to address this problem: (i) To reject the idea of uniform reference configuration and describe all fields relative to a non-uniform shape. (ii) To assemble geometrically incompatible elementary volumes into a connected region by immersing them into a space with extra features, characterizing the nonEuclidean properties. In this book, we consider only simple materials [104]. Their response is fully defined by linear transformation, determined by local deformation. Thus, each REV can be transformed to a uniform state by means of a linear transformation; these transformations taken together define a continuous field. At the same time, the geometry of the space with affine connection can be determined by Cartans’ moving frame method, which also uses the field of linear transformations of coordinate basis vectors. Thus, the construction of a region containing arbitrary distorted elementary volumes with mutually compatible shapes, gives use to the mathematical construction of a space with a general affine connection. The specific geometric properties of the thus-built space, i.e., the inner geometry of the body, characterize the measure of incompatibility and sources of residual stresses. In order to explain such argumentation, we shall give the simple example of a deformable membrane, assuming that the images of the reference and the actual shapes are known, such that one of them, or both, cannot be immersed into a Euclidean plane. We assume that the observers, describing the “physical reality,” are fictional “twodimensional creatures” populating the membrane, a two-dimensional surface, like inhabitants of Sphereland in Sphereland: a fantasy about curved spaces, written by D. J. Burger [107]. They cannot directly observe the deformation of the membrane, but

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121

feel changes in the physical properties of their two-dimensional world, caused by deformation. In addition to “two-dimensional observers,” deformation of the membrane is followed by powerful “three-dimensional observers,” who can see the shape of the membrane in its possible embedding in the enveloping three-dimensional space. Let us consider two situations. An image of a uniform configuration of the membrane cannot be embedded into a Euclidean plane, but it can be embedded in an enveloping Euclidean space as part of the sphere (Figure 4.1). In such an embedding, small neighborhoods are in a uniform, or physically indistinguishable, state. Some of them are shown as circumferences in Figure 4.1. “Two-dimensional observers” drawing on measurements of infinitesimal geometrical objects cannot say anything about the “spherical” shape of their world and may be convinced that their world is flat. Suppose now that the physical space is flat. Any embedding of the membrane in the physical space has to be accompanied by a deformation. In this case, two-dimensional observers will find out that their uniform world is no longer uniform: circumferences, characterizing the neighborhoods in Figure 4.1, are distorted, and such distortion occurs without any visible reasons for the two-dimensional observers, because no external force fields have been applied. The distorted circumferences are shown in Figure 4.1 by the solid lines. For comparison, undistorted circumferences are shown nearby by the dotted lines. According to [8], the membrane acquired material inhomogeneity, remaining materially uniform, i.e., made from one and the same material. It is clear that changes of geometric images can be observed only by the powerful “three-dimensional observers.” Images in Figure 4.1 are shown from their point of view. Two-dimensional observers will notice only changes in the physical properties of their world. For example, the rigidity of the membrane changes ununiformally, while if the membrane possesses optical properties, it becomes optically inhomogeneous; a sort of a rainbow will appear in the world of twodimensional observers.

Fig. 4.1: Embedding of a two-dimensional body with a non-Euclidean inner geometry in a flat physical space.

122 | 4 Geometric Formalization of the Body and Its Representation in Physical Space Remark 4.1. The embedding of a two-dimensional body S into an ambient three-dimensional space E, shown in Figure 4.1, is defined by using the formulae for a stereographic projection: X=

2x , 1 + x2 + y2

Y=

2y , 1 + x2 + y2

Z=−

x2 + y2 − 1 . x2 + y2 + 1

(4.1)

Here, x, y are the surface coordinates that can be perceived by a two-dimensional observer, and X, Y, Z are the Cartesian coordinates of the enveloping Euclidean space, defining the geometrical positions for the points of a two-dimensional embedded body that can be perceived by powerful three-dimensional creatures. The circumferences, symbolizing uniform neighborhoods of the points on S for three-dimensional observers are represented by the parametric formulae X = X0 +

r0 n3 cos φ , ω

Y = Y0 +

r0 n3 sin φ , ω

Z = Z0 +

ω = √n23 + (n1 cos φ + n2 sin φ)2 ,

r0 (n1 cos φ + n2 sin φ) , ω

φ ∈ [0, 2π[ ,

where X0 , Y0 , Z0 are three-dimensional coordinates of the centers of circumferences, calculated by (4.1) for the specified x0 , y0 , and r0 is their radius (identical for all), and (n1 , n2 , n3 ) are the components of the normal vector to the body image. The embedding of the body S in the two-dimensional physical space P is given by the transformation X Y x̃ = , ̃y = . 1+Z 1+Z As a result of this transformation, the circumferences are transformed into closed curves, parameterized in P as follows: (x, y) 󳨃→ (̃x , ̃y), ωX0 + r0 n3 cos φ , ω + ωZ0 − r0 (n1 cos φ + n2 sin φ) ωY0 + r0 n3 sin φ ̃y = , ω + ωZ0 − r0 (n1 cos φ + n2 sin φ) ̃x =

where φ ∈ [0, 2π[. The circumferences, symbolizing uniform neighborhoods, are transformed into distorted objects from the point of view of two-dimensional observers for reasons that are not related to the visible deformation of their two-dimensional world, whereas a three-dimensional observer visualizes this deformation explicitly. However, two-dimensional observers may assume that, for some reason, their two-dimensional world changes its geometric properties: its spherical geometry has been transformed to a flat geometry. Both points of view can be used to describe the embedding of the body in the physical space; however, appealing to a powerful “multidimensional observer” is often cumbersome and related to difficult-to-interpret geometric images. In the present

4.1 Geometric Motivation

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Fig. 4.2: Embedding of a two-dimensional body with a Euclidean inner geometry in a non-Euclidean physical space.

book, we develop methods of the “internal observer,” who can use two notions: deformation of the body and change of the geometry of the space, containing this body. In the example discussed, the space containing a uniform reference of the body shape has a non-Euclidian structure, while the physical space is a Euclidean plane. It is easy to imagine an inverted situation: a uniform shape is defined by flat geometry and the physical space is a Riemannian space with non-trivial curvature. Figure 4.2 illustrates this argument: a flat membrane is deposited such that it can slip onto the non-deformable spherical foundation. The images were also built by using transformation (4.1). Thus, incompatible deformations of the body are characterized by two geometrical concepts: the embedding in the physical space and the structure of the space containing its reference uniform configuration. It is possible to obtain another geometric interpretation of a non-Euclidean structure of the reference shape. Let the body B and the physical space P be presented by two-dimensional manifolds, where, by the condition, P is Euclidean. It can be shown by two orthogonal axes (x1 , x2 ) as in Figure 4.3. Assuming that the physical space of P is associated with the observer A, who “sees” everything happening in P, but is unable to see what is beyond it, the observer lives in a flat world like the characters of Flatland, written by E. A. Abbott [108]. We assume that the physical space of P represents a typical layer of the three-dimensional space of events S = 𝕋 × P, in which the added dimension corresponds to time; motions in P are characterized by changes of the relative positions of points in P when the layer P is transferred along the added chronometric axis 𝕋. In the space S, there exists a more powerful observer B, who sees all images of configurations of the body B in total, i.e., observes the world tube of the body B (Figure 4.3). Each section of the global tube is an image of the configuration, corresponding to a certain time instant. If the body has a natural configuration,

124 | 4 Geometric Formalization of the Body and Its Representation in Physical Space

(a)

(b) Fig. 4.3: A uniform reference configuration fully (a) and partially embedded (b) into the physical space of the observer.

the observer A can see this configuration in one of the sections. However, in a general case, such a section may not be present; the observer A concludes that deformations of the body B are incompatible. At the same time, the observer A may detect the sequence of sections (from his perspective, the sequence of time instants), in which the material points in specific curvilinear segments are in the natural state. Unlike observer A, a more powerful observer B sees all these segments in total and observes some two-dimensional surface in S, the intersection of the general surface and the global tube of the body. All material points of this surface are in a uniform state and, therefore, the surface represents an image of the natural configuration of the body. Thus, the observer A sees the image of a uniform configuration part by part, whereas the observer B sees the entire image. It is important that the image of the configuration for compatible and incompatible deformations, a two-dimensional surface and its “visualization,” depends on the geometrical fantasy of the observer. The fact that the segments are “glued together” in the section of the global tube, which can be formalized by a smooth manifold, is based on the additional hypothesis on

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the smoothness of a uniform configuration. We can imagine a more general situation, when at every time instant the neighborhood of one point transforms to a uniform state. In this case, the observer B sees a “thread” passing along the global tube.

4.2 Comparison between Conventional and Non-Euclidean Continuum Mechanics The most significant aspects that need to be changed or clarified in the transition from the classical Cauchy theory of deformations (Euclidean deformations) to generalizations to manifolds with non-Euclidean connection are given in Tab. 4.1. Here are the physical and geometric explanations: (i) The affine structure of the physical space E allows one to determine the positions of points by means of the field of the placement vector P 󳨃→ r = P − O. In the case of arbitrary smooth manifolds, the notion of the placement vector becomes meaningless; the position of the points is determined by local coordinates (an analogous situation arises in theoretical physics in the transition from the special theory of relativity to the general theory of relativity). (ii) In Euclidean continuum mechanics, the infinitesimal material fiber is represented by the difference in the vectors of the places of its beginning and end. In the non-Euclidean case, the infinitesimal material fiber associated with the point X of the body B is represented by the class of curves equivalent at this point, i.e., the tangent vector to the smooth manifold B at the point X. The set of vectors tangent to the smooth manifold B at the point X is endowed with the Tab. 4.1: The correspondence between notions of Euclidean and non-Euclidean theories of deformation. Notions

Euclidean continuum mechanics

Non-Euclidean continuum mechanics

(i)

Place

Local coordinates (x k )

(ii) (iii)

Infinitesimal material fibers Deformation

(iv) (v) (vi) (vii) (viii) (ix)

Configuration Displacement vector Deformation gradient Forces Tensors Fields

Vector field of places r (in Euclidean sense) ∆r = r2 − r 1 , where r 1 , r 2 are placement vectors γ : E ⊃ S R → S t ⊂ E, or, since E ≅ V, V ⊃ U R → U t ⊂ V Not used u=x−X F = 1 + (∇u)T , where ∇ = e k ∂ k Vectors V ⊗ V, V ⊗ V ⊗ V, . . . E →V⊗V

Equivalence class of curves P ⊃ 𝜘R (B) → 𝜘t (B) ⊂ P 𝜘: B → P Not used F = T𝜘 Covectors ∗ TX B ⊗ TX B, TX B ⊗ TX B, . . . Sections

126 | 4 Geometric Formalization of the Body and Its Representation in Physical Space structure of a vector space of dimension n = dim B and is denoted by TX B. This is the tangent space to the manifold B at the point X. Remark 4.2. To distinguish the tangent vectors from the manifolds from the vectors of the translation space V, the latter are denoted by lowercase bold Latin characters u, v, . . . , and the tangent vectors are denoted by lowercase Latin characters u, v, . . . . (iii) In the Euclidean continuum mechanics, it is assumed that there is a global natural (stress free) reference shape S R ⊂ E, and the concept of the body is not used. Deformations are represented by mappings of the form E ⊃ S R → S t ⊂ E, where S t is the actual shape. By virtue of the isomorphism E ≅ V, deformations can be represented in an equivalent way as mappings of subsets of V, i.e., as the maps V ⊃ U R → U t ⊂ V. In the non-Euclidean case, the deformation is the mapping¹ P ⊃ 𝜘R (B) → 𝜘 t (B) ⊂ P. Here, 𝜘 R (B), unlike in the classical case, can be a stressed shape. (iv) The concept of configuration is absent in the Euclidean theory, since the body is identified with the natural reference shape. At the same time, the role of the “deformation” from the imaginary space of affine connection, that is the superstructure over the body, in which an unstressed (or, more generally, uniform) shape is considered, to the actual shape of the physical space is played by configuration. (v) The affine structure of space in Euclidean theory allows one to introduce the displacement vector field u = x − X, the value of which represents the difference of the places of the particle occupied by it in the actual and reference shapes. In the case when the physical space is an abstract Riemannian manifold, the concept of a vector displacement field is absent. (vi) In Euclidean theory, the deformation gradient is determined by the field of linear transformations (the mappings X 󳨃→ F X = γ󸀠 (X), each of which, F X , linearizes the deformation γ in the neighborhood of the point X). Equivalently, the deformation gradient is represented in the form F = 1 + (∇u)T , where ∇u denotes the gradient of the vector field (in Gibbs notation [46]). In the approach developed, the deformation gradient is determined from the configuration 𝜘 : B → P and is represented by the tangent mapping F = T𝜘 : TB → TP (see Chapter 13, Section 13.2.2 and Section 13.3.6). (vii) In contrast to classical representations, it seems expedient to distinguish between vectors and covectors on a smooth manifold. We denote by T ∗p M the cotangent space to the smooth manifold M (representing the body or physical space), that is, the space of linear functionals over T p M. From the physical point of view, the elements of the dual space are the forces [35].

1 Here, 𝜘R is the reference configuration, and 𝜘t is the actual one.

4.2 Comparison between Conventional and Non-Euclidean Continuum Mechanics | 127

(viii) In classical mechanics, tensors and tensor fields are determined in the framework of the axioms of Euclidean space. At the same time, to work with fields in a non-Euclidean space, one needs a definition of a tensor that does not use the axioms of Euclidean space. (ix) A tensor field on M (including vector and covector) is considered as a section of the vector bundle constructed over M (see Chapter 13, Section 13.3). The coordinate frame on dim M = s is denoted by the symbol (∂ i )si=1 , and the coframe corresponding to it is denoted by (dx i )si=1 (see Chapter 13, Section 13.3.8). Thus, for an arbitrary smooth vector u : M → TM and covector ν : M → T ∗ M fields one has decompositions: u = u i ∂ i , ν = ν i dx i . Remark 4.3. The use of vector bundles allows one to consider sections as mappings of one smooth manifold into another, which, in turn, allows one to determine the smoothness of the sections in the standard way (see Chapter 13, Section 13.1.3). Remark 4.4. Tensors and tensor fields in Euclidean space are denoted by bold uppercase Latin symbols T, F, . . . , and tensor fields on smooth manifolds are handwritten Latin symbols T, F, . . . . The exception is E and V, which are reserved for affine Euclidean space and the associated vector space. (x)

The Euclidean structure of space provides the observer with an abstract “instrument”, which makes it possible to measure the lengths of vectors (infinitesimal fibers) and the angles between them. The presence of such a “device” is necessary for the construction of the theory of deformations. In the case of smooth manifolds, the role of the instrument is played by the Riemannian metric (see Chapter 13, Section 13.3.10). The latter exists on any smooth manifold. Using the Riemannian metric, one can introduce musical isomorphisms that associate a vector with a covector (operation (⋅)♭ ) and vice versa (operation (⋅)♯ ). Note that from a physical point of view, musical isomorphisms allow measuring physical quantities in units of measurement unusual for them, for example, force (covector) in meters, and displacement (vector) in Newtons. In index notation, these operations correspond to the lowering and raising of indices. (xi) Often, there is a need for the expression of tensor fields given at points of the actual shape through the points of the reference shape, and vice versa. In classical mechanics, these operations correspond to the Piola transformation. In the case of smooth manifolds, the analog of the Piola transform is pullback and pushforward operations. If T is a tensor field on P, and Q is a tensor field on B, then the pullback of T is denoted by 𝜘∗ T, and through 𝜘∗Q the pushforward of Q is denoted. Here, 𝜘 : B → P is a smooth map. In the case of a scalar function, its pullback and pushforward are reduced to replacing its domain. For an arbitrary tensor field, this is not so; except for the replacement of the domain, it is necessary to replace the tangent and cotangent bundles with special recalculation formulae (see Chapter 13, Section 13.3.7).

128 | 4 Geometric Formalization of the Body and Its Representation in Physical Space

4.3 A Body A body B is a set, whose elements are denoted by fraktur characters, X, Y, Z, . . . , and are referred to as material points. Its cardinal number, Card(B), defines the type of mechanical theory, which describes the kinematics and dynamics of material points that constitute the body. In the framework of analytical mechanics, finite or countable sets of material points are taken into account, i.e., Card(B) ≤ Card(ℕ) = ℵ0 . The subject of research in continuum mechanics is bodies with cardinal number Card(B) = Card(ℝ) = ℵ1 . Therefore, we will treat a body as a set with continuum cardinality. In order to describe interaction between the subsets of material points one needs to formalize the notion of a neighborhood. Such a notion can be formalized in the framework of topology. That is, one needs to assume that the body is a topological space. From the physical standpoint, different points should have disjoint neighborhoods, such as the space should be Hausdorff. For an analytical description of physical fields that are acting on material points, one requires to cover the body with charts (U, φ). Such charts locally introduce coordinates on the body B; each chart (U, φ) consists of an open set U ⊂ B and a homeomorphism φ : U → O between domain U of the chart and an open set O ⊂ ℝn . The number n is fixed. In the general case, such a covering can be provided by not less than a continuum of charts. The assumption that B has a countable base allows one to deal with finite or countable numbers of charts. To deal with smooth fields only, an additional requirement is needed: if domains U and V of charts (U, φ) and (V, ψ) intersect (U ∩ V ≠ 0), then the composition, ψ ∘ φ−1 |φ(U∩V), that acts between open sets in ℝn , is a C∞ -diffeomorphism. Thus, a smooth atlas on B should be chosen. One considers subsets (parts) of the body and provides integration over them, or their boundaries, with the purpose to formulate balance laws. The integration can be performed correctly in the case of oriented manifolds. Thus, one needs to introduce an orientation on B. This orientation is induced on all parts of the body. All above reasonings lead to the following properties, which the set B of material points is required to satisfy: (B1 ) Card(B) = ℵ1 ; (B2 ) B is an n-dimensional C∞ -manifold². (B3 ) B is orientable manifold, i.e., a volume form μ0 ∈ Ω n (B) exists for it. Suppose that an orientation [μ0 ] was chosen. By a part of the body B we mean an ndimensional embedded submanifold P ⊂ B with boundary and orientation [ι∗P μ 0 ]. The set of all parts of the body is denoted by Part(B). The representation of a body as a smooth manifold was used for the first time by W. Noll, C.-C. Wang, M. Gurtin, and A. Murdoch in [104, 109, 110]. For instance,

2 Brief information about the smooth manifold notion is given in Chapter 13, Section 13.1.

4.3 A Body | 129

Noll introduced an axiomatic definition of a smooth body as a set, equipped with a family of its mappings in the physical space called configurations and satisfying some special system of axioms. Gurtin and Murdoch gave a self-contained definition of a two-dimensional material surface based on the Noll axioms. Remark 4.5. The Noll system of axioms is based on the notion of the configuration 𝜘 as a mapping of the set B, representing a collection of material points constituting the body, to the absolute (Euclidean) physical space E, i.e., 𝜘 : B → E. Let us assume that a class of configurations C is given. The body B is called a continuous body of the class C p , if the class of configurations C satisfies the following conditions [104]: (C1 ) Each configuration 𝜘 ∈ C is a homeomorphism, and the image 𝜘(B) is an open set in E. (C2 ) If γ, 𝜘 ∈ C, then the composition³ λ = γ∘̂ 𝜘−1 : 𝜘(B) → γ(B) represents a mapping of the class C p , which is called deformation of the body B from the configuration 𝜘 to the configuration γ. (C3 ) If 𝜘 ∈ C and if λ : 𝜘(B) → E is a mapping of the class C p , then λ ∘ 𝜘 ∈ C. Thus, the order of body smoothness is defined by the smoothness of the compositions of its configurations. According to such a definition, the smooth body is a smooth manifold with trivial topological structure, i.e., a trivial manifold, whose atlas contains only one chart [3]. Such manifolds do not cover the possible variants of bodies (the need for using non-trivial manifolds, representing the body, is also described in [109]). M. Gurtin and A. Murdoch in [110] represent the material surface (a two-dimensional body) as a body, such that each image of a configuration into Euclidean affine space is a surface. Each surface Ω in E is a subset in E, such that a pair (T p Ω, π p ) is assigned to each point p ∈ Ω. Here, T p Ω is a two-dimensional vector subspace of V, and π p is a mapping. The family {(T p Ω, π p )}p∈Ω satisfies the following axioms: (Ω1 ) For each point p ∈ Ω, the mapping π p : Np → Ω is of class C2 . Its domain, Np , is an open neighborhood of 0 in T p Ω. (Ω2 ) For each p ∈ Ω, there exists a neighborhood U(p) of the point p ∈ Ω in E, such that π p (Np ) ∩ U(p) = Ω ∩ U(p). (Ω 3 ) For each p ∈ Ω, the vector π p (h) − (p + h) belongs to (T p Ω)⊥ for each h ∈ Np and is of order o(‖h‖) as ‖h‖ → 0; (Ω4 ) There exists a continuous field m : Ω → V, which never vanishes and has m(P) ∈ (T P Ω)⊥ for each p ∈ Ω. Here, (T p Ω)⊥ is the subspace of V, which is orthogonal to T p Ω: (T p Ω)⊥ = {v ∈ V | ∀u ∈ T p Ω : (u ⋅ v = 0)} . In [110], the set T p Ω is referred as a tangent space at point p ∈ Ω. Such a definition of the tangent space essentially uses the Euclidean and affine structure of E. Since

̂ : B → 𝜘(B) is obtained from 𝜘: B → E by restricting its codomain to range. 3 The mapping 𝜘

130 | 4 Geometric Formalization of the Body and Its Representation in Physical Space

we intend to consider bodies in Riemannian physical space, it is preferable to use the intrinsic definition of tangent space. Consider an example. Let B be a two-dimensional body that can be presented by a single sphere in a three-dimensional Euclidean space; its image is described by the equation x21 + x22 + x23 = 1, where x1 , x2 , x3 are the Cartesian coordinates. On this body, we can define an atlas that consists of two charts (U1 , φ1 ), (U2 , φ2 ), whose coordinate homeomorphisms are defined as stereographic projections. Thus, stereographic projections of the parts of the sphere on the plane, the points of which have a zero coordinate x3 when this plane is identified with ℝ2 , are presented by the following mappings: x1 x2 h1 : B \ (0, 0, −1) ∋ (x1 , x2 , x3 ) 󳨃→ ( , ) , 1 + x3 1 + x3 x1 x2 , ) . h2 : B \ (0, 0, 1) ∋ (x1 , x2 , x3 ) 󳨃→ ( 1 − x3 1 − x3 the first of these establishes a one-to-one correspondence between the points of the sphere and the plane, “pierced” by a beam beginning in the “southern pole” point with the coordinates (0, 0, −1), and the second establishes a one-to-one correspondence between the points of the sphere and the plane, “pierced” by a beam beginning in the “northern pole” point with the coordinates (0, 0, 1). In Figure 4.4 the dashed line with the sign of embedding designates the relation between sets and subsets, and the arrow designates the direction of action of the mapping. −1 The pre-images h−1 1 (D), h 2 (D) of the open disk D = {(ξ 1 , ξ 2 ) | (ξ 1 )2 + (ξ 2 )2 < 74 } −1 are used as U1 and U2 . The restriction of the mappings h−1 1 and h 2 on D are taken as φ1 : D → U1 and φ2 : D → U2 :

φ1 : (ξ 1 , ξ 2 ) 󳨃→ ( φ2 : (η1 , η2 ) 󳨃→ (

2ξ 1 2ξ 2 ξ1 + ξ2 − 1 , , ); 1 + (ξ 1 )2 + (ξ 2 )2 1 + (ξ 1 )2 + (ξ 2 )2 1 + (ξ 1 )2 + (ξ 2 )2

2η1 2η2 η1 + η2 − 1 , , − ) . 1 + (η1 )2 + (η2 )2 1 + (η1 )2 + (η2 )2 1 + (η1 )2 + (η2 )2

To the region U12 = U1 ∩ U2 in ℝ2 there corresponds a set of ordered pairs (ξ 1 , ξ 2 ), such that 47 < (ξ 1 )2 + (ξ 2 )2 < 74 . In this case, the transition homeomorphism is as follows: φ12 : U12 → U12 ,

(η 1 , η2 ) 󳨃→ (ξ 1 , ξ 2 ) = (

(η1 )2 (η2 )2 , ) , (η1 )2 + (η2 )2 (η1 )2 + (η2 )2

and is a C∞ -diffeomorphism. Thus, the sphere B is a two-dimensional manifold with the C∞ -structure induced by the constructed atlas (Figure 4.4).

4.3 A Body | 131

Fig. 4.4: Charting of the sphere and the transition function between the charts.

Since the body B is a smooth manifold, there exists a Riemannian metric G ∈ Sec(T ∗ B ⊗ T ∗ B). We refer to G as the material metric. Such a metric is not arbitrary; it is specific to considering physical theory. In Chapter 8, Section 8.4, the example of constructing the material metric for materially uniform simple bodies is shown. In another chapters where the material metric is involved, we assume that it is obtained from reasonings that are not dependent on those of such chapters. Note that unlike the case of the body, the spatial Riemannian metric g of the physical space P is pre-defined in our considerations. In a more general situation, both G and g should be obtained from certain reasonings. In this case, the body B and the physical space P are equal in rights. Such a consideration is beyond the scope of our book. The dimensions of the body and the physical space satisfy the following inequalities: n = dim B ≤ dim P = m ≤ 3 .

132 | 4 Geometric Formalization of the Body and Its Representation in Physical Space

4.4 Configurations Let B be a body and P be a physical space with the Riemannian metric g. The relation between the continual set of material points that constitute the body and the set of their places in the physical space is established by means of a configuration, that is, a map 𝜘 : B → P. The image of a configuration 𝜘, i.e., the set 𝜘(B), is referred to as a shape and is denoted by S𝜘 , if the body is fixed. The relations between a body, a configuration, and the shape are illustrated in the following diagram:

The diagram uses the following designations: ∙ ιS𝜘 : S𝜘 → P is the inclusion map. ̂ : B → S𝜘 is the map that is obtained from 𝜘 : B → P by restricting its codomain ∙ 𝜘 to S𝜘 . ̂. ∙ 𝜘 = ιS𝜘 ∘ 𝜘 Since both the body and the physical space are smooth manifolds, we take from the set of all configurations 𝜘 only those that are of class C∞ , that is, 𝜘 ∈ C∞ (B; P). This allows one to use differential calculus for further derivations. Additionally, 𝜘 is assumed to be a smooth immersion; at each material point X ∈ B the tangent map TX 𝜘 : TX B → T𝜘(X) P is injective. As we will discuss in Section 5.5, the map TX 𝜘 represents the configuration gradient at X, and thus the assumption states that spatial images of infinitesimal material fibers do not coincide. This condition is also referred to as the local impenetrability condition. Such an assumption does not guarantee that

4.4 Configurations

|

133

Fig. 4.5: Möbius strip.

Fig. 4.6: Klein bottle.

this impenetrability condition is global. The class of all smooth immersions 𝜘 must be restricted by the provision that 𝜘 is a homeomorphism onto its image 𝜘(B) ⊂ P in ̂ is a homeomorphism. Thus, to satisfy this the subspace topology⁴, or, equivalently, 𝜘 requirement 𝜘 needs to be a smooth embedding. Consider two examples to illustrate the difference between smooth immersion and smooth embedding, represented by the Möbius strip (Figure 4.5) and the Klein bottle (Figure 4.6), which can be considered as two-dimensional bodies. For the Möbius strip, one can define the configuration with the image in three-dimensional Euclidean space E: φ φ φ p(r, φ) = (3 + r cos ) cos φi1 + (3 + r cos ) sin φi2 + r sin i3 , 2 2 2 (r, φ) ∈ ] − 1, 1[ × [0, 2π[. Here (i 1 , i2 , i 3 ) is an orthonormal frame, and p is the position vector field. This set has no self-intersections and thus the corresponding configuration is a smooth embedding. At the same time, the projection of such a set on any plane, for example, X1 OX2 , is not a smooth embedding on the two-dimensional Euclidean space. There is a possibility to construct an embedding of the Möbius strip in a two-dimensional space that is topologically equivalent to the Klein bottle. If the physical space P is represented by a smooth manifold with such a structure, then for bodies that are topologically equivalent to the Möbius strip, there exists a continual family of configurations as smooth embeddings. This example provides an analogy to geometric continuum mechanics. If a body has internal stresses in Euclidean physical space, then the idea is to construct its embedding on some curved (non-Euclidean) space, where it will have a stress-free shape.

4 If SP is a topology on P, then the set SS𝜘 = {O ∩ S𝜘 | O ∈ SP } is the subspace topology of S𝜘 .

134 | 4 Geometric Formalization of the Body and Its Representation in Physical Space The Klein bottle can be represented in three-dimensional Euclidean space E as the image of the mapping (t, θ) 󳨃→ p(t, θ), p(t, θ) = α(t) + r(t) (cos θ G(τ(t)) + sin θi 3 ) ,

(t, θ) ∈ ]0, 2π[ × [0, 2π[ ,

where α(t) = a(1−cos t)i 1 +b sin t(1−cos t)i 2 , r(t) = c−d(t−π)√ t(2π − t), G(xi 1 +yi2 ) = −yi1 + xi2 , τ(t) = ‖α 󸀠 (t)‖−1 α 󸀠 (t). Such a mapping is a smooth immersion but with self-intersections. The minimal (relative to a dimension) space, into which the Klein bottle can be embedded, is ℝ4 . This corresponds to the results of the theory of smooth manifolds, in which theorems on embeddings are proved. One of them, the “weak Whitney theorem,” asserts that for any compact n-dimensional C r -manifold M, where r ≥ 2, there exists a C r -embedding of M into ℝ2n+1 . The proof can be found in [111, 112]. To strengthen the theorem, see the remark in [112, p. 27]. In particular, the “strong Whitney theorem” asserts that any smooth n-dimensional (n > 1) manifold can be immersed into ℝ2n−1 and can be embedded into ℝ2n . In this case, the values of 2n − 1 and 2n are unimprovable. The different ways of Klein bottle parametrization are represented in [113]. The surface, which was illustrated in Figure 4.6, was obtained by the following parameter values: (a, b, c, d) = (20, 8, 11/2, 2/5). Remark 4.6. The necessity of carrying out research on configurations that are immersions is obvious for bodies whose dimension is less than the dimension of the physical space. However, such configurations can be of interest also in the case when the dimensions of the body and physical space coincide. The “Möbius crystal”, investigated by Wang [109], is an example. Let us briefly consider the example. The body is materially uniform [104, 109] if all its points are mutually materially isomorphic. Physically, this means that all its material points consist of the same material. Let the physical space be a three-dimensional Euclidean space E and a stress-free shape of a body B represented by a long circular cylinder. Assume that the material is anisotropic and has a rhombic symmetry. This means that three crystallographic axes (e 1 , e2 , e3 ) are associated with each point of the body. These axes are such that the mechanical properties of the material do not change as the shape rotates through an angle π around each of these axes. There are no other material symmetries. Assume that in the stress-free shape, all material points are oriented in such a way that the corresponding crystallographic axes are parallel, and one of them, for example e1 , is parallel to the axis of the cylinder. In such a shape, there exists a unique smooth material isomorphism [109], represented by a translation in Euclidean space. We make the following transformations: first we tighten the cylinder so that its lower base rotates relative to the upper one by an angle π, then we bend the cylinder into the ring, and “glue” the bases. C.-C. Wang called the resulting solid of the toroidal form the “Möbius crystal”. It is clear that any simply-connected part of the body thus obtained has a smooth field of material isomorphisms, but it cannot be extended to the whole body.

4.5 A Shape of a Body as a Submanifold of the Physical Space |

135

In the rest of the book, by a configuration of a body B in the physical space P we mean a C∞ -embedding 𝜘 : B → P. The set of all such mappings is denoted by C(B; P).

4.5 A Shape of a Body as a Submanifold of the Physical Space 4.5.1 A Shape as a Submanifold Let B be an n-dimensional body, P be a physical space with the Riemannian metric g, and 𝜘 ∈ C(B; P) be some configuration. The shape S𝜘 ⊂ P is endowed with subspace topology. Since 𝜘 is a smooth embedding, S𝜘 is an embedded n-dimensional submanifold⁵ of P. That is, by means of 𝜘, S𝜘 is endowed with a C∞ -structure such that the inclusion map ιS𝜘 : S𝜘 → P is a C∞ -embedding. Such a structure is constructed as follows. If AB = {(U α , φ α )}α∈I is some smooth atlas from the smooth structure of the body B, then the set ̂ −1 |𝜘(U α ) )} A𝜘 = {(𝜘(U α ), φ α ∘ 𝜘 α∈I is a smooth atlas on S𝜘 . It generates the desired smooth structure of S𝜘 . The construĉ −1 |𝜘(U α ) ) is illustrated in the following diagram: tion of a chart (𝜘(U α ), φ α ∘ 𝜘 𝜘

B

𝜘̂

ι Uα

𝜘α

Uα

𝜘̂ α

φα ℝn

̃ φα

P ιS𝜘 S𝜘 ι𝜘(U α ) 𝜘(U α )

The diagram uses the following designations: ̂ on U α ⊂ B, i.e., 𝜘α = 𝜘 ̂ ∘ ι Uα . ∙ 𝜘 α is the restriction of 𝜘 ̂ α obtained from 𝜘 ̂ by restricting of its domain and codomain on U α and 𝜘(U α ), ∙ 𝜘 ̂α = 𝜘 ̂ ∘ ι Uα . i.e., ι𝜘(U α ) ∘ 𝜘 −1 ̃ ̂ α is the coordinate homeomorphism. ∙ φα = φα ∘ 𝜘

5 See Chapter 13, Section 13.1.4 for more detail.

136 | 4 Geometric Formalization of the Body and Its Representation in Physical Space 4.5.2 The Local k-slice Condition One can define a smooth atlas on the shape S𝜘 by means of a smooth atlas of the body B. Meanwhile, as was stated above, only objects that belong to the physical space P (curves, tangent vectors, scalar functions, etc.) are observable. If an observer introduces some coordinate system on P, then he intends to use such a system for constructing a smooth atlas of S𝜘 . This can be formalized by the local k-slice condition [3]. Let us consider this formalization in more detail. Let M be a smooth n-dimensional manifold and (U, φ) a smooth chart on M. A subset S ⊂ U is a k-slice of U, k ∈ {0, . . . , n}, if φ(S) ⊂ ℝn has the form φ(S) = {(x1 , . . . , x k , x k+1 , . . . , x n ) ∈ ℝn | x k+1 = ⋅ ⋅ ⋅ = x n = 0} . Here, (x i ) are local coordinates that are generated by the coordinate homeomorphism φ. By the definition, the last n − k coordinates of each point from S are equal to zero. In accordance with [3], S satisfies the local k-slice condition if for each point p of S there exists a smooth chart (U, φ) on M, such that p ∈ U and U ∩ S is a k-slice of U. The theorem [3] asserts that: if S ⊂ M is an embedded k-dimensional submanifold of M, then S satisfies the local k-slice condition. Therefore, there exists a family {(U p , φ p )}p∈S of charts on M that cover S and such that for any p ∈ S, the chart domain of (U p , φ p ) contains p, and S ∩ U p is a k-slice of U p . The family⁶ A S = {(S ∩ U p , π ∘ φ|S∩U p )}p∈S constitutes a smooth atlas of S. These reasonings are illustrated in the diagram S ∩ Up

ι U p ∩S

(x1 , . . . , x k , 0, . . . , 0) ℝk

Up (x1 , . . . , x n )

π

ℝn

Here, π : ℝn → ℝk is the projection onto the first k coordinates, that is, π : (x1 , . . . , x k , x k+1 , . . . , x n ) 󳨃→ (x1 , . . . , x k ) .

6 In [3] it is proven that such atlas generates a smooth structure that is the unique one in which S can be considered as a submanifold.

4.5 A Shape of a Body as a Submanifold of the Physical Space |

137

4.5.3 The Induced Riemannian Space Structure Let B be a n-dimensional smooth body and P be an m-dimensional physical space with a Riemannian metric g. The Riemannian space structure of P can be induced on ∗ . The diagram each shape S𝜘 by means of the pullback ιS 𝜘 P

π1

∗ ιS 𝜘

ιS𝜘 S𝜘

T∗P ⊗ T∗P

π2

T ∗ S𝜘 ⊗ T ∗ S𝜘

∗ illustrates the action of ιS ; here π1 and π2 are the projections from the corresponding 𝜘 total spaces of vector bundles onto their bases. Thus, since g ∈ Sec(T ∗ P ⊗ T ∗ P) is the Riemannian metric on P, the section ∗ g ∈ Sec (T ∗ S𝜘 ⊗ T ∗ S𝜘 ) ιS 𝜘

is the Riemannian metric on S𝜘 . It acts in the following way: ∗ g(u, v) := g(TιS𝜘 ⌞ u, TιS𝜘 ⌞ v) . ∀u, v ∈ TS𝜘 : ιS 𝜘

(4.2)

Here, TιS𝜘 : TS𝜘 → TP is the tangent map of the inclusion map. If (s α )nα=1 are local coordinates on S𝜘 that are induced from the local k-slice condition, and (x i )m i=1 are local coordinates on P, then T x ιS𝜘 is represented by the dyadic decomposition T x ιS𝜘 =

󵄨 ∂x i 󵄨󵄨󵄨 󵄨 ∂ i |x ⊗ ds α |x , α ∂s 󵄨󵄨󵄨󵄨φ(x) x

where x i : (s1 , . . . , s n ) 󳨃→ x i (s1 , . . . , s n ), i = 1, . . . , m, is the coordinate representation of ιS𝜘 in a neighborhood of x, and φ is the coordinate homeomorphism that generates (s α )nα=1 . Thus, by (4.2) one has ∗ ιS g = g ij 𝜘

∂x i ∂x j α ds ⊗ ds β . ∂s α ∂s β

(4.3)

Here, g ij = g(∂ x i , ∂ x j ).

4.5.4 A Shape and the Physical Space. Intrinsic versus Spatial Let the physical space be the classical three-dimensional Euclidean space E. Imagine that there are two bodies, B1 and B2 , where the first one is three-dimensional and the second one is two-dimensional. Suppose that these bodies are in contact, that is, in

138 | 4 Geometric Formalization of the Body and Its Representation in Physical Space some configurations 𝜘1 and 𝜘2 one has that the surface 𝜘2 (B2 ) is a part of the boundary ∂𝜘1 (B1 ) for an open set 𝜘1 (B1 ). The stress and strain states in B1 are defined via three-dimensional tensor fields that have values in the tensor product V ⊗ V ∗ of the associated vector space V. Since B2 is two-dimensional, one requires clarifying the notion of stress and strain on such a body. This can be done in the framework of the theory developed by Gurtin and Murdoch in [110]. One obtains two-dimensional tensor fields that cannot be set into any correspondence with three-dimensional ones without additional geometrical considerations. In particular, this consideration defines the operation In[. . .] that allows one to transform the surface divergence div s of the two-dimensional Cauchy stress tensor field σ s into a three-dimensional vector field and thus to write the balance equality div T + b = In[divs σ s ] . Here, T is the Cauchy stress tensor in B1 and b is a body force. Let us consider this problem in more detail. Let P be a physical space and S𝜘 be some shape. Vector bundle structures over S𝜘 are defined such that they are compatible with features of pure geometrical objects, that “populate” the manifold, figuratively speaking, its “inhabitants”, such as smooth curves γ : 𝕀 → S𝜘 and smooth functions f : S𝜘 → ℝ. Similarly, vector bundle structures over P can also be defined by smooth curves γ : 𝕀 → P and smooth functions f : P → ℝ. These structures are not related directly. For example, a smooth section u ∈ Vec(S𝜘 ) takes values in TS𝜘 , but not in TP. This means that the observer in P cannot measure values of u without using extra data. This data is provided by the tangent map TιS𝜘 . Denote this map by InS𝜘 . That is, InS𝜘 := TιS𝜘 : TS𝜘 → TP . Applied to the aforementioned section u, such a map gives a new one, v = InS𝜘 ⌞ u : S𝜘 → TP. The new field v takes values in TP and, thus, these values can be measured by the observer. We see that InS𝜘 plays the role of the additional geometric data, which aligns tangent vectors from TS𝜘 with spatial tangent vectors from TP. Note that such a property was implicitly used in (4.2). Let us consider the correspondence between tangent vectors on shape and spatial ones in more detail. The restrictions of InS𝜘 on tangent fibers, i.e., the maps InS𝜘 ;x := T x ιS𝜘 : T x S𝜘 → T x P ,

(4.4)

x ∈ S𝜘 , are injective and linear, with rank(InS𝜘 ;x ) = dim B ≤ dim P. This means that for any x ∈ S𝜘 , the map InS𝜘 ;x is an isomorphism onto its image. Thus, one has the opportunity to identify vectors from T x S𝜘 with the corresponding vectors from the n-dimensional subspace of T x P. Neglecting such an opportunity results in more detailed, but more complicated, theory. As a rule, we use such identification only in the case when the body and the physical space have the same dimensions, i.e.,

4.5 A Shape of a Body as a Submanifold of the Physical Space |

139

dim B = dim P. That is, if dim B = dim P, then we identify linear spaces T x S𝜘 and T x P. Each tangent vector u ∈ T x S𝜘 is identified with the corresponding spatial tangent vector InS𝜘 ;x (u) ∈ T x P. Note that when dim B = dim P, the shape S𝜘 is an open subset of P. This corresponds to the classical considerations of continuum mechanics in which shapes are defined as open sets with boundaries that are regular in the sense of Kellogg.

5 Strain Measures 5.1 Review on Cauchy Theory The geometric ideas for describing the small change in the shape and volume of bodies under the action of external fields applied to them go back to the works of Bernoulli and Euler, but the geometric relations of the theory of finite deformations were obtained by Augustin-Louis Cauchy and are given in most modern monographs on continuum mechanics without significant changes [59]. Despite the fact that Cauchy’s arguments are inextricably linked with the postulates of Euclidean space, the Cauchy method is easily carried over to spaces of a more general structure, in particular, Riemannian ones. In this regard, it seems appropriate to comment on the provisions it adopts (the authors used the works [73, 78, 114]). i) The existence of a natural shape. The concept of the body is not used by Cauchy to describe deformations, it is sufficient to consider the body shapes, among which, as expected, there is a special, natural stress-free one. It is in this shape that there is a body in the absence of external fields, and the application leads to distortion of the shape and deformation. ii) Euclidean structure of the physical space. Appealing to the classical analytic geometry of Descartes, Cauchy identifies points in a distorted shape by their Cartesian coordinates x, y, z. Note that Cauchy used a distorted shape to identify points, because it is observable. iii) The existence of displacements. The Cauchy deformation is considered as a process of “returning” a shape to an undistorted state (in modern terminology this is an inverse deformation [115]), which is realized for the shape as a whole and is characterized by a continuous displacement field with components ̃ξ : (x, y, z) → ̃ξ (x, y, z) ,

̃ : (x, y, z) → η ̃ (x, y, z) , η

̃ζ : (x, y, z) → ̃ζ (x, y, z)

in the Cartesian frame [78]. Thus, “incompatible deformations” are excluded. iv) The physical state of the body depends on the distortions of elementary material fibers. Cauchy postulates this implicitly, considering as arguments of all field values the ratio of the lengths of infinitesimal material segments with their distortion in the composition of the body shape. To calculate the lengths, the Cartesian metric generated by the relations i p ⋅ i s = δ ps , where (i s )3s=1 is Cartesian frame, is used. Remark 5.1. An infinitely small material segment was determined by the coordinates of the particles (molecules, in the interpretation of the physics of the nineteenth century) located at its ends. In the actual form, these material particles correspond to the Cartesian coordinates x, y, z and x + ∆x, y + ∆y, z + ∆z. The coordinates of the particles https://doi.org/10.1515/9783110563214-005

142 | 5 Strain Measures

in the reference shape were represented by expressions [78] x − ξ, y − η, z − ζ ,

and

x + ∆x − ξ − ∆ξ, y + ∆y − η − ∆η, z + ∆z − ζ − ∆ζ ,

̃ (x, y, z), ζ = ̃ζ (x, y, z). where ξ = ξ̃ (x, y, z), η = η To quantify the distortion of material fibers, Cauchy calculates their lengths r and r/(1 + ε) in distorted and undistorted shapes: r2 = (∆x)2 + (∆y)2 + (∆z)2 ,

(

2 r ) = (∆x − ∆ξ)2 + (∆y − ∆η)2 + (∆z − ∆ζ)2 , 1+ε

and, taking into account their infinitesimality, decomposes the last relation by the Taylor formula of the first order [78]: (

2 2 r ∂ξ ∂ξ ∂ξ ∆x − ∆y − ∆z − ⋅ ⋅ ⋅ ) ) = (∆x − 1+ε ∂x ∂y ∂z

+ (∆y −

2 ∂η ∂η ∂η ∆x − ∆y − ∆z − ⋅ ⋅ ⋅ ) ∂x ∂y ∂z

+ (∆z −

2 ∂ζ ∂ζ ∂ζ ∆x − ∆y − ∆z − ⋅ ⋅ ⋅ ) . ∂x ∂y ∂z

(5.1)

In the context of modern terminology, this means that the study material is simple [37, p. 60]. Expression (5.1) is characterized by a quadratic form corresponding to an ellipsoid, a triple of mutually orthogonal axes and three numbers corresponding to the main semi-axes. This Cauchy ellipsoid assumes the main characteristic of the deformed state at a point. Since these constructions implicitly exclude rotations accompanying deformations, one can see in them a principle that, within the framework of modern terminology, is called the principle of objectivity or the principle of material indifference¹. For the sake of justice, it should be noted that Cauchy did not complete the construction of the theory of finite deformations, and the famous Cauchy polar decomposition, which explicitly distinguishes the rotations accompanying deformations was not strictly proved [59], despite the fact that he carried out this research for about 10 years. Here, however, it should be taken into account that Cauchy did not have a modern theory of operators in finite-dimensional spaces and carried out all constructions in terms of semiaxes of ellipsoids. v) Duality of distorted and undistorted shapes. In the work [114], unlike in [78], to identify the Cauchy points, their coordinates are used in an undistorted shape, thereby emphasizing the duality of distorted and undistorted shapes. Historically, earlier references to this duality are available in the works of J. Lagrange [117].

1 Note that the need to use the principle of material indifference was pointed out by Poisson, who first drew attention to the fact that rigid stresses in the body should not change in tension [116]. From the point of view of the general theory of relativity, the principle of material indifference is a special case of the principle of general covariance [11].

5.2 Configurations and Deformations

| 143

In fact, there is no reason to prefer one of the shapes over another, and the equations characterizing the deformation can be written both relative to the reference and relative to the actual shapes. Mathematically, the transition from one description to another is carried out on the basis of the formulae for the transformation of areas and volumes in Euclidean space (Nanson’s formula [37, 124]) and leads to the well-known relations of Piola [37, p. 124]. The duality of the relations obtained in this case is shown in [115]. The generalization of the positions i)–v) and the transfer of the generalization to structurally inhomogeneous bodies, i.e., manifolds with non-Euclidean geometry, reduces to a modification of the procedure for comparing infinitesimal material fibers associated with different points of the body with a uniform (reference) fiber. Geometrically, this means the definition of the non-Euclidean material connection on the body [8]. For simple materials, it is an affine connection², and the basic relationships needed for such a generalization are similar in shape to the relations arising when using common curvilinear coordinates³, although, of course, their geometric meaning acquires a new content.

5.2 Configurations and Deformations Let B be an n-dimensional smooth manifold (dim B = n) that represents a body and let P be an m-dimensional smooth manifold (dim P = m) that represents a physical space. Suppose that P is endowed with the Riemannian metric g. Changing the shape of the body B, embedded in P, is quantified by deformation map, which is the subject of present section. In conventional continuum mechanics the pivot role in deformation quantification belongs to the analysis of shapes of material fibers, which change in the course of deformation. In the framework of the geometrical approach, material fibers can be defined as smooth curves χ : 𝕀 → B on the body. Such curves are observable only if some configuration 𝜘 ∈ C(B; P) is chosen. That is, one observes not the set χ(𝕀) of material points, but the set 𝜘 ∘ χ(𝕀) of their places. Let 𝜘R be a configuration that the observer considers as a reference. Thus, 𝜘 R ∘ χ(𝕀) is the reference image of the material fiber. If 𝜘 is a configuration, which image, 𝜘(B), the observer considers as being now, as actual, the set 𝜘 ∘ χ(𝕀) is the actual image

2 By the method of the moving frame in space one can introduce the affine connection. To do this, one uses a smooth field of non-degenerate linear transformations. Taking into account that the response of simple materials is completely determined by the smooth field of linear transformations, W. Noll constructed an affine connection on the body with the help of the moving frame method, transforming it into a uniform shape [104]. 3 Apparently, the brothers E. Cosserat and F. Cosserrat [118] began to use systematically curvilinear coordinates in the theory of elasticity.

144 | 5 Strain Measures

of the same material fiber. Using the measurement tool provided by the Riemannian metric g, one can calculate the lengths of reference and actual shapes of the material fiber. Therefore, it is possible to compare metric properties of the reference and actual images of the fiber. To this end, define the map γ : 𝜘 R (B) → 𝜘(B) ,

̂∘𝜘 ̂ −1 γ := 𝜘 R ,

which is referred to as a deformation of B from the configuration 𝜘 R into the configuration 𝜘 [104]. Relations between the configurations and deformations are illustrated in the following diagram: B

ιS

𝜘 P

𝜘̂

P

𝜘̂R

𝜘R

ι S𝜘

𝜘 R

S𝜘R

γ

S𝜘

In the diagram, S𝜘R = 𝜘 R (B) and S𝜘 = 𝜘(B). The artistic form of this diagram is shown in Figure 5.1.

Fig. 5.1: Configurations and deformations.

5.3 Coordinate Representations of Configurations and Deformations 5.3.1 The General Case Since the body B and the physical space P are represented by smooth manifolds with dim B = n and dim P = m, one can take some smooth atlases AB = {(U α , φ α )}α∈I , and

5.3 Coordinate Representations of Configurations and Deformations

| 145

AP = {(V β , ψ β )}β∈J from their C∞ -structures. Let 𝜘, 𝜘 R ∈ C(B; P) be configurations. Since S𝜘R and S𝜘 are embedded n-dimensional submanifolds of P, one can choose smooth atlases AS𝜘R = {(W δR , σ R;δ )}δ∈K R and AS𝜘 = {(W τ , σ τ )}τ∈K from their smooth structures. Fix some material point X ∈ B. Denote X = 𝜘 R (X) and x = 𝜘(X). The points X and x determine the spatial places of the material point X. Obtain coordinate representations of the configurations 𝜘 R and 𝜘 in some chart domain U α , α ∈ I, such that X ∈ U α . Choose indices β 1 , β 2 ∈ J, such that X ∈ V β1 and x ∈ V β2 . Then, denoting −1 U αβ1 β2 := U α ∩ 𝜘−1 R (V β 1 ) ∩ 𝜘 (V β 2 ), one arrives at the following coordinate representations for the configurations 𝜘 R and 𝜘: ̃ R;α,β1 ,β2 = ψ β1 ∘ 𝜘 R ∘ φ−1 𝜘 α : φ α (U αβ 1 β 2 ) → ψ β 1 (V β 1 ) , ̃ α,β1 ,β2 = ψ β2 ∘ 𝜘 ∘ φ−1 𝜘 α : φ α (U αβ 1 β 2 ) → ψ β 2 (V β 2 ) . These mappings act on open subsets of ℝn and take values in ℝm . Obtain coordinate ̂ R and 𝜘 ̂ . Choose such indices δ ∈ K R representations for their restrictions on shapes, 𝜘 R ̂ −1 ̂ −1 (W τ ), one and τ ∈ K that X ∈ W δR and x ∈ W τ . Denoting U αδτ := U α ∩ 𝜘 R (W δ ) ∩ 𝜘 obtains the following coordinate representations: R ̃̂ R;α,δ,τ = σ R;δ ∘ 𝜘 ̂ R ∘ φ−1 𝜘 α : φ α (U αδτ ) → σ R;δ (W δ ) ,

̃̂ α,δ,τ = σ τ ∘ 𝜘 ̂ ∘ φ−1 𝜘 α : φ α (U αδτ ) → σ τ (W τ ) . Such mappings act on open subsets of ℝn into ℝn . Finally, one arrives at the coordinate −1 ̂∘𝜘 ̂ −1 representations for deformation γ = 𝜘 R and its inverse, γ , in neighborhoods of X and x: ̃γ τ,δ = σ τ ∘ γ ∘ σ −1 R;δ : σ R;δ (S R;αδτ ) → σ τ (W τ ) , R ̃ γ−1 δ,τ = σ R;δ ∘ γ−1 ∘ σ −1 τ : σ τ (S αδτ ) → σ R;δ (W δ ) .

Here, S R;αδτ = 𝜘 R (U αδτ ) and S αδτ = 𝜘(U αδτ ). Such mappings act between open sets in ℝn . Relations between restricted configurations, deformations, and their coordinate representations are illustrated in the diagram ℝn

⊃

φα U αδτ φα ℝn

⊃

̃̂ R;α,δ,τ 𝜘

φ α (U αδτ )

φ α (U αδτ )

σ R;δ

𝜘̂ R

𝜘̂

σ R;δ (S R;αδτ )

⊂

ℝn

S R;αδτ ̃γ τ,δ

γ S αδτ ̃̂ α,δ,τ 𝜘

στ σ τ (S αδτ )

⊂

ℝn

146 | 5 Strain Measures

5.3.2 The Euclidean Case In the particular case when P is represented by m-dimensional affine-Euclidean space min E with associated vector space V, one can choose the minimal atlas AE . Such an atlas consists of one chart (E, D), in which D is the mapping (2.2), i.e., ∀a ∈ E :

D(a) := ((a − o) ⋅ i k )m k=1 ,

where (o; (i k )m k=1 ) is a fixed orthonormal coordinate system on E. One may add other min charts to the minimal atlas AE , which are C∞ -compatible with the original chart (E, D). This way is equivalent to introducing curvilinear coordinates on E. Consider coordinate representations of configurations and deformations in the Euclidean space E. We use two copies for ℝm : the space ℝm d , which contains Cartesian coordinates, and the space ℝm , which contains curvilinear ones. Choose some c point X ∈ B and pick up a chart (U α , φ α ) from the atlas AB , such that U α ∋ X. Let 𝜘 ∈ C(B; E) be a configuration. The affine structure of the physical space allows us to define the following mappings: 𝜘α = 𝜘 ∘ φ−1 α : Oα → E ; 𝜘α = p ∘ 𝜘 : O α → V ; 𝜘d;α = D ∘ 𝜘 : O α → ℝm d . Here, O α ⊂ ℝn is an open set, the codomain of φ α . The mapping 𝜘 α provides spatial representation of material points; their places in the physical space E are assigned to their local coordinates in the chart (U α , φ α ). The mapping 𝜘α provides the “vectorization” of the configuration; to each material point, which is represented by local coordinates in (U α , φ α ), its position relative to the fixed origin o is assigned by means of the position vector field. Finally, the mapping 𝜘d;α provides the “arithmetization” of the configuration; a m-tuple of the Cartesian coordinates is assigned to a place on each material point. Let us introduce curvilinear coordinates in E by means of some smooth atlas A c = min {(V β , ψ β )}β∈J , which is equivalent to AE . Let (V β , ψ β ) be any chart from such an atlas, for which X ∈ V β . The mapping n −1 m 𝜘c;α,β = ψ β ∘ 𝜘 ∘ φ−1 α : ℝ ⊃ φ α (U α ∩ 𝜘 (V β )) → ψ β (V β ∩ 𝜘(U α )) ⊂ ℝc ,

assigns to each point from a portion of U α the curvilinear coordinates of its place in E. The equality m h c;β = ψ β ∘ D−1 : ℝm d → ℝc defines a transformation of Cartesian coordinates into curvilinear ones. By means of such a mapping, one can obtain the following expression for 𝜘c;α,β: 𝜘 c;α,β = h c;β ∘ D ∘ 𝜘 ∘ φ−1 α ,

5.3 Coordinate Representations of Configurations and Deformations

| 147

and the formula that links 𝜘c;α,β and 𝜘 d;α : 𝜘 d;α = h−1 c;β ∘ 𝜘 c;α,β . To such a relation there corresponds the diagram (W αβ = U α ∩ 𝜘−1 (V β )): E

∪

∪ 𝜘

D

𝜘(W αβ )

h

W αβ

ℝm d

− c; 1 β

B

φα

ψβ

φ α (W αβ ) ∩ ℝn

𝜘 c;α,β

ψ β (𝜘(W αβ )) 𝜘 d;α ∩ ℝm c

̂∘𝜘 ̂ −1 Let 𝜘 R , 𝜘 ∈ C(B; E) be two configurations. Then, γ = 𝜘 R is the corresponding deformation. Since γ acts between submanifolds, one can apply the local k-slice condition, which allows one to obtain the corresponding atlases AS𝜘R = {(W δR , fR;δ )}δ∈K R and AS𝜘 = {(W τ , fτ )}τ∈K for S𝜘R and S𝜘 . Thus, the deformation has the following representation: −1 ̃γ τ,δ = fτ ∘ γ ∘ fR;δ : ℝn ⊃ Q αδτ → R αδτ ⊂ ℝn , where Q αδτ and R αδτ are open sets, which are defined similarly as in the preceding section. The attempt to represent the deformation γ in terms of D directly can fail in the case n < m, since one obtains a mapping that is not defined on some open subset of⁴ ℝm d . Such a mapping γ d cannot be differentiated by itself; for each point y of its domain D d , one needs to find an open neighborhood O d;y and such a smooth mapping Γ d;y defined on this neighborhood that its restriction to D d coincides with the coordinate representation of γ, i.e., Γ d;y |D d = γ d . m Consider the case n = m. Denote by D𝜘R : S𝜘R → ℝm d and D𝜘 : S𝜘 → ℝd the restrictions of D to the shapes S𝜘R and S𝜘 . Thus, {(S𝜘R , D𝜘R )} and {(S𝜘 , D𝜘 )} are the corresponding atlases. The coordinate representation of γ is the following: m γ d = D𝜘 ∘ γ ∘ D𝜘−1R : ℝm d ⊃ D ∘ 𝜘 R (U α ) → ℝd .

4 The set 𝜘R (U α ) is open in 𝜘R (B) but not open in E. Therefore, D ∘ 𝜘R (U α ) is not open in ℝm d.

148 | 5 Strain Measures Here, D ∘ 𝜘 R (U α ) is an open subset of ℝm d . The relation between γ and γ d is illustrated by the diagram E

ℝm d ∪

∪ 𝜘(U α )

B

φα

D ∘ 𝜘(U α )

𝜘̂

∪ Uα

D𝜘

γ

𝜘̂R

ℝm

𝜘 R (U α )

γd D𝜘R

D ∘ 𝜘 R (U α )

∩

∩

E

ℝm d

Now consider the representation of the deformation γ in the curvilinear coordinates. The atlas A c = {(V β , ψ β )}β∈J gives two atlases for the shapes S𝜘R and S𝜘 , which ̃ R;δ )}δ∈J and A c;𝜘 = {(S𝜘 ∩ V β , ψ ̃ β )}β∈J . In these are denoted by A c;𝜘R = {(S𝜘R ∩ V δ , ψ 𝜘R 𝜘R ̃ R;δ and ψ ̃ β are restrictions of the original coordinate atlases, coordinate mappings ψ mappings from the atlas A c . Choose δ ∈ J 𝜘R , β ∈ J 𝜘 , such that 𝜘 R (X) ∈ V δ and 𝜘(X) ∈ V β . Denoting U αβδ = U α ∩ 𝜘−1 (V β ) ∩ 𝜘−1 R (V δ ) one arrives at the coordinate representation ̃β ∘ γ ∘ ψ ̃ −1 : ℝm ⊃ ψ δ ∘ 𝜘 R (U αβδ ) → ψ β (S𝜘 ∩ V β ) ⊂ ℝm . γc = ψ c c R;δ The relation between γ and γ c is shown in the diagram E

ℝm c ∪

∪ 𝜘(U αβδ )

B ∪

ℝm

ψ β ∘ 𝜘(U αβδ )

𝜘 γ

U αβδ φα

̃β ψ

γc

𝜘R 𝜘 R (U αβδ )

̃ R;δ ψ

ψ δ ∘ 𝜘 R (U αβδ )

∩

∩

E

ℝm c

5.4 Two-Point Tensors |

149

5.4 Two-Point Tensors 5.4.1 Two-Point Tensor Bundle The deformation gradient F and the first Piola–Kirchhoff stress tensor P that arose in conventional elasticity (Chapter 3) are examples of two-point tensors. To generalize these notions on smooth manifolds one requires some generalization for the notion of the tensor bundle. Such a generalization comes down to the notion of a two-point tensor bundle, which is introduced in the present section. Let (E, N, π) be a smooth real vector bundle, M be a smooth manifold, and f ∈ C∞ (M; N) be some smooth mapping. The mapping f induces the new vector bundle (f ∗ E, M, π∗ ) upon the original one, (E, N, π). The vector bundle (f ∗ E, M, π∗ ) is referred to as the pullback bundle [119]. The total space of such a bundle is represented by the set f ∗ E := {(p, x) ∈ M × E | f(p) = π(x)} = ⋃ {p} × Ef(p) , p∈M

and the projection the form

π∗ :

f ∗E

→ M is defined via

π ∗ (p, x)

:= p. Thus, fiber over p has

(f ∗ E)p = {p} × Ef(p) . The notion of the pullback bundle allows one to define a two-point tensor bundle. If M and N are smooth manifolds, f ∈ C∞ (M; N), then⁵ T (r,s) f ∗ TN ⊗ T (k,l) TM , is the total space of such a bundle. Its base is M and fiber over p is the following: {p} × (T (r,s)(T f(p) N) ⊗ T (k,l) (T p M)) . Each element of T (r,s) f ∗ TN ⊗ T (k,l) TM is referred to as a two-point tensor over the map f [60, 120, 121]. Choose some open set V ⊂ N and let U = f −1 (V). Assume that (E α ), (E α ) are the local frame and coframe for TM, defined on U, while (e i ), (e i ) are the local frame and coframe for TN, defined on V. By e i 1 ⊗ ⋅ ⋅ ⋅ ⊗ e i r ⊗ e j1 ⊗ ⋅ ⋅ ⋅ ⊗ e j s ⊗ E α1 ⊗ ⋅ ⋅ ⋅ ⊗ E α k ⊗ E β 1 ⊗ ⋅ ⋅ ⋅ ⊗ E β l , we denote local section, which to each point p ∈ U assigns the mixed tensor p 󳨃→ e i1 |q ⊗ ⋅ ⋅ ⋅ ⊗ e i r |q ⊗ e j1 |q ⊗ ⋅ ⋅ ⋅ ⊗ e j s |q ⊗ E α1 |p ⊗ ⋅ ⋅ ⋅ ⊗ E α k |p ⊗ E β1 |p ⊗ ⋅ ⋅ ⋅ ⊗ E β l |p ,

5 One can define bundles of another form, like T (k,l) TM ⊗ T (r,s) f ∗ TN, etc.

150 | 5 Strain Measures where q = f(p). The family of all such sections constitutes the local frame for T (r,s) f ∗ TN ⊗ T (k,l) TM. In this book, we use second-order two-point tensors. All possible types of such tensors are listed below: T ∈ Sec(f ∗ TN ⊗ T ∗ M) ,

T : p 󳨃→ T iα (p)e i |f(p) ⊗ E α |p ,

T ∈ Sec(f ∗ TN ⊗ TM) ,

T : p 󳨃→ T iα (p)e i |f(p) ⊗ E α |p ,

T ∈ Sec(f ∗ T ∗ N ⊗ T ∗ M) ,

T : p 󳨃→ T iα (p)e i |f(p) ⊗ E α |p ,

T ∈ Sec(f ∗ T ∗ N ⊗ TM) ,

T : p 󳨃→ T i α (p)e i |f(p) ⊗ E α |p .

(5.2)

Consider the transformation laws of such tensors. Assume that primed frames (E󸀠α ), (e󸀠i ) and not primed frames (E α ), (e i ) are related via equalities, E󸀠α = Ω α E β , β

e󸀠i = ω i e j , j

where Ω : p 󳨃→ Ω(p) ∈ GL(dim M; ℝ) and ω : p 󳨃→ ω(p) ∈ GL(dim N; ℝ) are smooth local fields of invertible matrices defined on some open subsets M and N, respectively. Then, for coframes one has α

E󸀠 = (Ω−1 )α β E β ,

i

e󸀠 = (ω−1 )i j e j .

Thus, the following transformation laws for the tensors (5.2) take place: T󸀠

i

= (ω−1 )i j Ω α T β , β

α

j

T 󸀠 iα = ω i Ω α T jβ , j

β

T󸀠

iα

= (ω−1 )i j (Ω−1 )α β T jβ ,

α

T 󸀠 i = ω i (Ω−1 )α β T j . β

j

There is a close relationship between second-order two-point tensors and vector bundle homomorphisms [3]. The following commutative diagrams illustrate actions of such homomorphisms: TM

T

π TM M

TM

π TN f T

π TM M

TN

M

T∗N

T∗M

N

T

f T

π T∗M M

TN π TN

π T∗M

N

π T∗N f

T∗M

N

T∗N π T∗N

f

N

By the definition, for each mapping T from such diagrams the restriction to each fiber is linear. There is the following one-to-one correspondence between such mappings

5.4 Two-Point Tensors

|

151

and second-order two-point tensors: T : TM → TN

↔

T ∈ Sec(f ∗ TN ⊗ T ∗ M) ,

T : T ∗ M → TN

↔

T ∈ Sec(f ∗ TN ⊗ TM) ,

T : TM → T ∗ N

↔

T ∈ Sec(f ∗ T ∗ N ⊗ T ∗ M) ,

T : T∗M → T∗N

↔

T ∈ Sec(f ∗ T ∗ N ⊗ TM) .

5.4.2 The Transpose and Orthogonal Tensors The transpose and orthogonal tensors are heavily used in continuum mechanics. One of the many examples is the Cauchy polar decomposition results in Cauchy–Green strain measures. In the conventional finite-strain theory, the transpose F T to the deformation gradient F : SR → Lin(V; V) is defined as a unique field F T : SR → Lin(V; V) that satisfies the property (3.11), ∀u, v ∈ V : v ⋅ F X [u] = u ⋅ F TX [v] , for all X ∈ SR . The orthogonal tensor in Euclidean space is defined by (12.5), i.e., it is a linear mapping Q ∈ Lin(V; V) that preserves the inner product: ∀u, v ∈ V : Q[u] ⋅ Q[v] = u ⋅ v . Each of these definitions essentially uses Euclidean properties of the associated vector space V. In the case of smooth manifolds, instead of Cartesian inner product one can use either the contraction ⟨⋅, ⋅⟩ or the Riemannian metric g (if it is fixed). Since no affine structure is presumed, one needs to consider four possible two-point tensors (5.2). This leads to several definitions of the transpose and orthogonal tensors [60, 121]. In the book, we use the definitions of the transpose tensor for the first one in (5.2). These definitions agree with those in [121] but have some technical changes. In the article referred to, the mapping f : M → N (or, in terms of the article, φ : B → I) is assumed to be a bijective. Such an assumption is strong enough, since we apply the theory developed to the case when f is a configuration, i.e., f is bijective onto its image only. Let f ∈ C ∞ (M; N). Consider the pullback bundle ι∗f(M) TN. Its base is the set f(M) and the fiber over each point p ∈ f(M) is represented by the set {p} × T p N. Such a vector bundle represents the restriction of TN to f(M). Assume that smooth manifolds M and N are equipped with Riemannian metrics G and g. For T ∈ Sec(f ∗ TN ⊗ T ∗ M) we define its transpose as the two-point tensor T T ∈ Sec ((̂f −1 )∗ TM ⊗ ι∗f(M) T ∗ N) ,

152 | 5 Strain Measures such that for each x ∈ f(M) the mapping TxT ∈ Lin(T x N; T y M), where y = ̂f −1 (x), satisfies the condition ∀u ∈ T y M ∀v ∈ T x N :

gx (Ty u, v) = Gy (u, TxT v) .

(5.3)

The two-point tensor T T can be considered as the vector bundle homomorphism T T : ι∗f(M) TN → TM . Note that such a mapping is defined not on the total space of the tangent bundle (Tf(M), f(M), π), but on the total space of the vector bundle (ι∗f(M) TN, f(M), π ∗ ). The relation between the mappings T T and ̂f is illustrated on the diagram T ι∗f(M) TN T

π∗ f(M)

TM π TM

̂f −1

M

The formula for the transpose tensor components follows directly from (5.3). If T = T iα e i ⊗ E α , then T T has the following dyadic decomposition: T T = (T T )α i E α ⊗ e i , where (T T )α i = T T (E α , e i ). According to (5.3), in which u = E α , v = e j , one obtains, β

(G)αβ (T T ) j = g ij T iα . Finally, one arrives at the formula j

(T T )α i = g ij G αβ T β ,

(5.4)

where G αβ = G(E α , E β ) and g ij = g(e i , e j ). Note that this formula corresponds to the classical one, (3.15), obtained in Chapter 3. Another definition of the transpose does not require the metrics. We use the pullback bundle ι∗f(M) T ∗ N. Its base is f(M) and the fiber over each point p ∈ f(M) is represented by the set {p} × T ∗p N. For T ∈ Sec(f ∗ TN ⊗ T ∗ M), we define its conjugate as the two-point tensor T ∗ ∈ Sec ((̂f −1 )∗ T ∗ M ⊗ ι∗f(M) TN) , such that for each x ∈ f(M) the mapping Tx∗ ∈ Lin(T ∗x N; T ∗y M), where y = ̂f −1 (x), satisfies the condition ∀u ∈ T y M ∀ν ∈ T ∗x N :

⟨Ty u, ν⟩x = ⟨u, Tx∗ ν⟩ y .

(5.5)

5.5 Configuration Gradient

| 153

The physical interpretation of the definition (5.5) is as follows. Such a definition determines the energy conjugacy: the power generated by the spatial density of forces ν on the physical field of the velocity Tu is equal to the power developed by the material density of forces T ∗ ν on the material velocity u. The two-point tensor T ∗ can be considered as the following vector bundle homomorphism: T ∗ : ι∗f(M) T ∗ N → T ∗ M . The relation between the mappings T ∗ and ̂f is shown on the diagram ∗ ι∗f(M) T ∗ N T T ∗ M

π∗

π T∗M

f(M)

̂f −1

M

If T = T i α e i ⊗ E α , then the formula (5.5) implies that T ∗ can be written as the following dyadic decomposition: T ∗ = T iα E α ⊗ e i . This means that the dyadic decomposition of T ∗ can be obtained from the original of T, just by interchanging its basis vectors in the dyad. The orthogonal tensor Q is a two-point tensor Q ∈ Sec(f ∗ TN ⊗ T ∗ M), such that ∀x ∈ M ∀u, v ∈ T x M :

Gx (u, v) = gf(x) (Qx u, Qx v) .

(5.6)

Using (5.3) one obtains Gx (u, v) = Gx (u, QTf(x)Qx v), for all u, v ∈ T x M. This leads to the relation Gx (u, (QTf(x)Qx − IdT x M )v) = 0, which holds for each u, v ∈ T x M. Thus, one has QTf(x)Qx = IdT x M , and we arrive at the conventional relation for the orthogonal tensor.

5.5 Configuration Gradient Any section u ∈ Vec(B) of the tangent bundle TB generates some material fiber χ : 𝕀 → B, such that u χ(t) = χ 󸀠 (t), for all t ∈ 𝕀. Figuratively speaking, the material fiber is “assembled” from the values of such a field. Conversely, for each material fiber there exists a smooth vector field with such a property. Thus, tangent vectors or, equivalently, derivations of the space of germs, can be viewed as infinitesimal elements of material fibers. In a fixed material point X ∈ B all such objects form the infinitesimal neighborhood of X, which is formalized by the tangent space TX B. Each element from TX B is the velocity at X for some material fiber that starts at X.

154 | 5 Strain Measures A configuration 𝜘 ∈ C(B; P) transforms material fibers into observable spatial fibers. The same role for infinitesimal objects is played by the configuration gradient F = T𝜘, which is the tangent bundle homomorphism, or, equivalently, the secondorder two-point tensor. That is, F : TB → TP ,

or ,

F ∈ Sec(𝜘∗ TP ⊗ T ∗ B) .

As follows from the first row of (5.2), F has the dyadic decomposition F = F i α e i ⊗ E α . Here, (E α ) is a local coframe on B, and (e i ) is a local frame on P. In particular, choosing coordinate frames, one obtains F=

∂𝜘 i (X1 , . . . , Xn ) ∂ i ⊗ dXα , ∂Xα

where 𝜘 i : (X1 , . . . , Xn ) 󳨃→ 𝜘 i (X1 , . . . , Xn ), i = 1, . . . , m, is the coordinate representation of the configuration 𝜘. If X ∈ B, then by FX we denote the corresponding linear map that acts between tangent fibers. That is, FX = TX 𝜘 ∈ Lin(TX B; T𝜘(X) P). ̂∘𝜘 ̂ −1 Let 𝜘 R , 𝜘 ∈ C(B; P) be two configurations and γ = 𝜘 R the corresponding deformation, then Tγ corresponds to the deformation gradient. Consider the classical case when P = E, i.e., when the physical space is represented by the affine-Euclidean space E with the translation vector space V. Let dim B = dimE = m. Compare the classical and smooth manifold approach for the notion of the deformation gradient. For a moment we will forget our convention about identifying of tangent spaces T x S𝜘 and T x P, which was stated in Section 4.5.4. According to conventional elasticity, the deformation gradient, F, is the total derivative of the deformation γ : S𝜘R → S𝜘 , which is defined on the open set S𝜘R , i.e., 󸀠 F = γ󸀠 . In an orthonormal frame (i k )m k=1 , the linear map F X = γ (X) has the following dyadic representation: 󵄨 ∂x k 󵄨󵄨󵄨 󵄨󵄨 ik ⊘ ij , (5.7) FX = ∂X j 󵄨󵄨󵄨(X 1 ,...,X m ) i m where (X i )m i=1 , (x )i=1 are Cartesian coordinates of points in the reference and actual shapes, (X 1 , . . . , X m ) 󳨃→ (x1 , . . . , x m ) = D ∘ γ ∘ D−1 (X 1 , . . . , X m ) is the coordinate representation of γ, and vectors i k are defined via relations i k ⋅ i s = δ ks .

Remark 5.2. In the framework of the present reasonings, for the dyadic representation in Euclidean space, we use the symbol ⊘ with the purpose of having a notation that is different from the general tensor product ⊗. In classical mechanics, the dyad u ⊘ v (linear map from V to V) is defined as follows: (u ⊘ v) ⋅ w := (v ⋅ w)u . The Cartesian coordinate map D can be restricted to each of the shapes S𝜘R , S𝜘 and to obtain atlases that are consist on one chart. One arrives at the following represen-

5.5 Configuration Gradient

tation: TX γ =

| 155

󵄨 ∂x k 󵄨󵄨󵄨 󵄨󵄨 ∂ X k |x ⊗ dX j |X . ∂X j 󵄨󵄨󵄨(X 1 ,...,X m )

In this case, InS𝜘R ;X = i k ⊗ dX k |X . Hence, −1 InS𝜘 ;x ∘ T X γ ∘ InS 𝜘

;X R

=

󵄨 󵄨 ∂x k 󵄨󵄨󵄨 ∂x k 󵄨󵄨󵄨 j 󵄨󵄨 󵄨󵄨 i ⊗ i ≈ ik ⊘ ij = F X . k ∂X j 󵄨󵄨󵄨(X 1 ,...,X m ) ∂X j 󵄨󵄨󵄨(X 1 ,...,X m )

(5.8)

On the left-hand side from the sign ≈, the element of the coframe (i k )m k=1 appears, and on the right-hand side, stands the element of the dual vectorial frame (i k )m k=1 . Such a distinction is the consequence of the ambient space Euclidean structure. When T X γ, considered as a bilinear form, acts on a vector, then such an action is realized via the operation ⌞: T X γ ⌞ u, which leads to contraction of such a vector with the corresponding covector in the dyadic product. The Euclidean structure is not used. On the contrary, according to classical continuum mechanics, the bilinear form F X , acts on a vector by means of (⋅) : F X ⋅ u. This leads to the scalar product of such a vector with the corresponding vector in the dyadic product. Choose local coordinates (Z i )m i=1 for the shape S𝜘R and “freeze” them into the shape. In a formal sense, this means that between coordinate mappings on S𝜘R and S𝜘 , the equality σ = σ R ∘ γ−1 holds. In this case, 󵄨 ∂X k 󵄨󵄨󵄨 󵄨󵄨 i k ⊗ dZ j |X , ∂Z j 󵄨󵄨󵄨(Z 1 ,...,Z m ) 󵄨 ∂x k 󵄨󵄨󵄨 󵄨󵄨 = i k ⊗ dZ j |x . ∂Z j 󵄨󵄨󵄨(Z 1 ,...,Z m )

InS𝜘R ;X = InS𝜘 ;x

Denote local frames on S𝜘R and S𝜘 by E k and e k : Ek =

∂X j ij , ∂Z k

Ek =

∂Z k j i , ∂X j

ek =

∂x j ij , ∂Z k

ek =

∂Z k j i . ∂x j

Thus, from the formula (5.7) and the chain rule, one obtains the classical formula for F X [45]: F X = e k |x ⊘ E k |X . The expression for T X γ takes the form T X γ = ∂ Z k |x ⊗ dZ k |X , since the coordinate representation ̃γ = σ ∘ γ ∘ σ −1 R is the identity map. In such a case, we obtain 󵄨 ∂Z k 󵄨󵄨󵄨 −1 󵄨󵄨 = e k |x ⊗ i j = e k |x ⊗ E k |X ≈ e k |x ⊘ E k |X = F X . (5.9) InS𝜘 ;x ∘ T X γ ∘ InS ;X 𝜘R ∂X j 󵄨󵄨󵄨(X 1 ,...,X m ) Here, i j ∈ V ∗ . Remark 5.3. In terms of the operation ⊗, the relations (5.8) and (5.9) between T X γ and F X are replaced by the formula −1 F X = InS𝜘 ;x ∘ T X γ ∘ InS 𝜘

R

;X

.

156 | 5 Strain Measures

5.6 Left and Right Cauchy–Green Strain Tensors 5.6.1 Spatial Measurements in Material Description The configuration gradient F provides the spatial description for an infinitesimal material fiber. Using the metrical data given by the Riemannian metric g, one can calculate the length of a material fiber in the spatial description and the angle between spatial realizations of two material fibers. At the same time, the length and angle can be considered in the material description. Assume that some material metric G in the body B is chosen. Let 𝜘 ∈ C(B; P) be some configuration and F = T𝜘 its configuration gradient. A vector u ∈ TB represents some infinitesimal material fiber, which spatial realization is v = F ⌞ u. The square of its length is given by g(v, v): g(v, v) = g(F ⌞ u, F ⌞ u) = G (u, FT ∘ F ⌞ u) , where (5.3) was used. That is, ‖v‖2g = G(u, C ⌞ u) ,

where ,

C = F T ∘ F : TB → TB .

(5.10)

Here, C is a vector bundle homomorphism over B; for each X ∈ B one has the symmetric linear mapping T CX = F𝜘(X) ∘ FX ∈ Lin(TX B; TX B) .

Assume that χ : ]a, b[ → B is some material fiber. Its spatial realization is represented by the smooth curve 𝜘 ∘ χ : ]a, b[ → B. By the spatial length of material fiber χ we mean the integral b

lg (χ) = ∫ √g𝜘∘χ(t)(Fχ(t) u(t), Fχ(t) u(t)) dt , a

where u(t) = χ 󸀠 (t) is the velocity vector. Using the formula (5.10) one obtains b

lg (χ) = ∫ √Gχ(t) (u(t), Cχ(t) u(t)) dt .

(5.11)

a

Finally, let χ 1 : ]a, b[→ B, χ2 : ]c, d[→ B be two material fibers, which intersect at X = χ 1 (t1 ) = χ2 (t2 ). Let u i : t 󳨃→ u i (t) = χ 󸀠i (t) be the velocity vector of χ i , i = 1, 2. The observable angle ∠(𝜘 ∘ χ 1 , 𝜘 ∘ χ2 )𝜘(X) between the fibers χ1 and χ2 at the observable point 𝜘(X) is the angle between their velocity vectors. That is, cos ∠(𝜘 ∘ χ2 , 𝜘 ∘ χ 1 )X =

gX (FX u 1 (t1 ), FX u 2 (t2 )) . √gX (FX u 1 (t1 ), FX u 1 (t1 ))√gX (FX u 2 (t2 ), FX u 2 (t2 ))

5.6 Left and Right Cauchy–Green Strain Tensors |

157

where X = 𝜘(X). Again using the relation (5.3) and the definition of C leads to the formula cos ∠(𝜘 ∘ χ 2 , 𝜘 ∘ χ 1 )X =

GX (u 1 (t1 ), CX u 2 (t2 )) . √GX (u 1 (t1 ), CX u 1 (t1 ))√GX (u 2 (t2 ), CX u 2 (t2 ))

(5.12)

Formulae (5.10), (5.11), and (5.12) provide the material description for the length and angle of the observed fibers. In the spatial description, one needs to know the configuration gradient F and the spatial metric g, while in the material description, one requires the metric G and the bundle homomorphism C. Consider the classical case of the Euclidean physical space. If the body is identified with some stress-free shape, then configurations are treated as deformations, and to the tensor C = F T ∘ F there corresponds a right Cauchy–Green strain tensor C = F T ∘ F. Thus, one can treat C as the generalization of the classical right Cauchy– Green strain tensor. For this reason, we refer to C as the right Cauchy–Green strain tensor as well.

5.6.2 The Cauchy Polar Decomposition Theorem In conventional considerations, right and left Cauchy–Green tensors appear by virtue of the Cauchy polar decomposition theorem. Such reasonings can be generalized on smooth manifolds [60]. For each X ∈ B, there exist orthogonal tensors⁶ RX : TX B → T𝜘(X) P, such that F =R∘U, FX = RX ∘ UX , F =V∘R,

FX = V𝜘(X) ∘ RX ,

where the symmetric⁷ positive-definite tensors UX : TX B → TX B and VX : T X P → T X P are, in conventional terminology, right and left stretch tensors. For simple bodies, the following combinations of the elements from (F, F T ) are used: the right C and left B Cauchy–Green strain tensors, which are vector bundle homomorphisms C = F T ∘ F : TB → TB , ∗ B = F ∘ F T : ιS TP → TP . 𝜘

In the corresponding tangent fibers, CX : TX B → TX B ,

CX = FXT ∘ FX ;

BX : T X P → T X P ,

BX = FX ∘ FXT ,

6 Orthogonal tensor was defined by (5.6). T 7 UX = UX and VX = VXT .

X = 𝜘(X) .

158 | 5 Strain Measures

The right and left stretch tensors are expressed via the Cauchy–Green strain tensors as follows: UX = √FXT ∘ FX = √CX , VX = √FX ∘ FXT = √BX .

5.6.3 Cauchy–Green Strain Measures as Pullback and Pushforward of Metrics Let X ∈ B, X = 𝜘(X). For any vector u ∈ TX B, the following relation holds: α

CX (u) = FXT (FX u) = (FXT ) i (FX )i β ∂Xα |X ⊗ dXβ |X ⌞ u . The equality (5.4) implies that j

CX = (GX )αγ (gX )ij (FX )i γ (FX ) β ∂Xα |X ⊗ dXβ |X . Finally, omitting the argument signs, one obtains C = G αγ g ij

∂𝜘 i ∂𝜘 j ∂Xα ⊗ dXβ . ∂Xγ ∂Xβ

(5.13)

Similarly, for any u ∈ T X P, one has α

BX (u) = FX (FXT u) = (FX )i α (FXT ) j ∂ X i |X ⊗ dX j |X ⌞ u . Using similar reasoning, one can obtain B = G αβ g lj

∂𝜘 l ∂𝜘 k ∂ X k ⊗ dX j . ∂Xβ ∂Xα

(5.14)

Denote C ♭b := C 1♭b . By the definition of (⋅)♭ , such a tensor is represented by the following dyadic representation: C ♭b = g ij

∂𝜘 i ∂𝜘 j dXα ⊗ dXβ . ∂Xα ∂Xβ

Thus, C is the pullback of the spatial metric: C ♭ b = 𝜘∗ g . An analogous formula holds for the pushforward of the material metric. One needs to ̂ = T̂ ̂ : B → 𝜘(B). The tangent map F use the mapping 𝜘 𝜘 : TB → T𝜘(B) serves as the ̂ The “internal” strain measure and is related with F = T𝜘 by the equality F = InS𝜘 ∘ F. ∗ T : TS → ̂ use of ιS g as the metric on S and the relation (5.3), allows one to define F 𝜘 𝜘 𝜘 ̂=F ̂∘F ̂T . TB. Thus, one can obtain the “internal” Cauchy–Green tensor B ̂2♯p . By the definition of (⋅)♯ , one has ̂♯p := B Set B 𝜘 k ∂̂ 𝜘l ̂♯p = G αβ ∂̂ ∂ k ⊗ ∂Xl . B α ∂X ∂Xβ X

5.6 Left and Right Cauchy–Green Strain Tensors

The last formula implies

| 159

∗

̂♯p = 𝜘 ̂ ∗ (G) , B ∗

where G := G1♯b 2♯b . Thus, the right Cauchy–Green tensor defines the pullback of the spatial metric, and the left one, the pushforward of the material metric. These reasonings are illustrated by the diagram C

B FX

TX P

TX B FXT C♭b

♭b

∗ B TX

♭p

FX∗

T ∗X P

B♯p

6 Motion 6.1 Motion as a Curve A motion of the body B in the physical space P can be described by a continual sequence of configurations. To such continual sequence there corresponds a continual family of shapes. Such a notion can be formalized as follows [60]. Let 𝕋 = ]t1 , t2 [ ⊂ ℝ be an interval of instants (t1 and t2 may be finite or infinite). The mapping M : 𝕋 → C(B; P), which assigns to each instant t ∈ 𝕋 some configuration 𝜘 t , i.e., M : t 󳨃→ 𝜘 t , represents a motion of the body B over the time interval 𝕋. That is, motion is a curve in C(B; P). Equivalently, the motion M can be represented as the family M = {𝜘t }t∈𝕋 of configurations. Any motion M = {𝜘t }t∈𝕋 defines two types of mappings. Let X ∈ B be a material point. Then, the mapping χ M;X : 𝕋 → P ,

χ M;X (t) := 𝜘 t (X) ,

represents a spatial trajectory of X. Another mapping is obtained by fixation of a time t ∈ 𝕋. That is, χ M;t : B → P , χ M;t (X) := 𝜘t (X) . Such a mapping is just the configuration 𝜘 t . If for any X ∈ B, the mapping χ M;X is of class C∞ , then we refer to M as a C∞ - (or smooth) motion. In this book, we consider only smooth motions. Fix a point X ∈ B. A smooth motion M = {𝜘 t }t∈𝕋 defines continual families of smooth charts¹ {(U t , φ t )}t∈𝕋 and {(V t , ψ t )}t∈𝕋 , such that ∀t ∈ 𝕋 : (X ∈ U t ) ∧ (𝜘t (U t ) ⊂ V t ) . Thus, for the point X one has 𝕋 ∋ t 󳨃→ 𝜘 t (X) ∈ ⋃ t∈𝕋 V t . The mappings χ M;X and χ M;t have the following coordinate representations: ̃χ M;X = ψ t ∘ χ M;X ∈ C∞ (𝕋; ℝm ) , ∞ n m ̃χ M;t = ψ t ∘ χ M;t ∘ φ−1 t ∈ C (ℝ ; ℝ ) .

1 Since one can pick up countable sets of charts that cover B and P, some elements of the families {(U t , φ t )}t∈𝕋 and {(V t , ψ t )}t∈𝕋 may coincide. Nevertheless, in our reasonings it is convenient to use index set 𝕋. https://doi.org/10.1515/9783110563214-006

162 | 6 Motion

The action of these mappings is illustrated in the diagram

X

∈

B

Vt

∪

∪

Ut

χ M;t

φt ℝn

𝜘 t (U t ) ψt

̃χ M;t

⊂

⋃t∈𝕋 V t

⊂ χ M;X

⊂

P

χ M;X 𝕋

̃χ M;X

ℝm

Note that in the framework of conventional continuum mechanics, the motion is considered as a family of deformations. Let 𝜘R be some chosen reference configuration. Thus, the motion is a family of transformations of the reference shape 𝜘 R (B) into actuals, 𝜘 t (B). Such transformations are represented by the family {γ t }t∈𝕋 of dêt ∘ 𝜘 ̂ −1 formations γ t = 𝜘 R . Relations between time-dependent configurations and deformations are illustrated in the diagram

It is noteworthy that in this pictorial diagram, the body is shown as a solid torus, while its shapes are depicted as “mugs” that are topologically equivalent to each other. As such, we wish to emphasize that the body and its shapes can be significantly different metric manifolds.

6.2 Velocity Let M = {𝜘t }t∈𝕋 be a C∞ -motion. Typically, there are two equivalent descriptions for kinematical and physical fields in continuum mechanics. The first one provides

6.2 Velocity

| 163

their definition as mappings with respect to positions of material points in some fixed shape. This is the material description. In this case, one observes the motion of individual particles. The second one involves the field determination as mappings with respect to fixed spatial points. This is the spatial description. The velocity fields in material and spatial descriptions are introduced as follows. Material description. The notion of the velocity vector to a curve, defined by the formula (13.18), being applied to the curve χ M;X gives two types of mappings. Fix a material point X ∈ B. Then, one has the mapping V M;X : 𝕋 → TP ,

V M;X (t) := χ󸀠M;X (t) ∈ T𝜘t (X) P .

Since the image of 𝕋 under mapping χ M;X is the spatial trajectory of the point X, which was observed at the instant interval 𝕋, the vector V M;X (t0 ), for t0 ∈ 𝕋 represents the velocity of the point X at some instant t0 . Thus, one has the field of velocities of a fixed material point. In contrast, one may be interested in velocities of all material points in a fixed instant t ∈ 𝕋. Let us define the mapping V M;t : B → TP ,

V M;t (X) := χ󸀠M;X (t) ∈ T𝜘t (X) P .

The value V M;t (X0 ) at X0 ∈ B represents the velocity of some point X0 at the instant t. Finally, one arrives at the assignment V : B × 𝕋 → TP ,

(X, t) 󳨃→ χ 󸀠M;X (t) .

which is called the material velocity field. For a material point X ∈ B and instant t0 ∈ 𝕋, one has the decomposition i V M;X (t0 ) = V M;X (t0 )∂ i |X , where X = 𝜘t 0 (X), i V M;X (t0 ) =

d[π im ∘ ψ ∘ χ M;X (τ)] 󵄨󵄨󵄨 , 󵄨󵄨 󵄨τ=t 0 dτ

i = 1, . . . , m .

Here, ψ is a coordinate homeomorphism with domain on P containing X. The material velocity V defines the family {V M;t }t∈𝕋 of mappings V M;t : B → TP. Each mapping V M;t assigns to every material point X ∈ B a tangent vector V M;t (X) ∈ T𝜘t (X) P. In this regard, V M;t is exactly a smooth section of the pullback bundle 𝜘∗t TP, i.e., V M;t ∈ Sec(𝜘∗t TP). Spatial description. Let t ∈ 𝕋. The spatial velocity is defined as the section of the ∗ pullback bundle ιS TP: 𝜘 t

̂ −1 v t = V(⋅, t) ∘ 𝜘 t : S𝜘t → TP . Finally, we define spatial velocity field as follows: v : ⋃ (S𝜘t × {t}) → TP , t∈𝕋

(X, t) 󳨃→ v t (X) .

164 | 6 Motion

6.3 (m + 1)-Formalism and Acceleration Acceleration fields are introduced in the framework of Newtonian space-time² (𝕋 × P, S, D, ∇(m+1) , pr1 ). The space-time manifold is represented by the (m + 1)-dimensional product manifold 𝕋 × P. The absolute time function is the projection pr1 : 𝕋 × P → 𝕋. The affine connection ∇(m+1) arises from the Levi–Civita connection ∇ on P with coefficients γ i jk . If we denote coefficients of ∇(m+1) by Γ ijk , then we have i

i

Γ jk

γ , { { { jk = {0, { { {0,

i, j, k ∈ {1, . . . , m} , (k = 0) ∧ (i, j ∈ {0, 1, . . . , m}) , (j = 0) ∧ (i, k ∈ {0, 1, . . . , m}) .

Let M = {𝜘 t }t∈𝕋 be a smooth motion. For a material point X ∈ B, its world line is represented by the curve³ (m+1)

χ M;X : 𝕋 → 𝕋 × P ,

(m+1)

χ M;X (t) := (t, χ M;X (t)) .

(m+1)

The (m + 1)-velocity V M;X of the point X is the field (m+1)

V M;X : 𝕋 → T(𝕋 × P) ,

(m+1)

(m+1)

V M;X (t) := (χ M;X )󸀠 (t) .

Taking the natural isomorphism T(t,X)(𝕋 × P) ≅ 𝕋 ⊕ T X P into account gives (m+1)

∀t ∈ 𝕋 : V M;X (t) = (1, V M;X (t)) . The (m + 1)-acceleration of the material point X is defined as follows: (m+1)

A M;X : 𝕋 → T(𝕋 × P) ,

(m+1)

(m+1) (m+1) (m+1) V M;X V M;X

A M;X = ∇

.

Like the (m + 1)-velocity, the (m + 1)-acceleration vector field splits into two fields as follows: (m+1) ∀t ∈ 𝕋 : A M;X (t) = (0, A M;X (t)) , where A M;X : 𝕋 → TP is the desired acceleration of the point X. Taking into account (m+1) the expressions for Γ ijk one obtains the following representation of A M;X in components: (m+1) 0 (A M;X ) = 0 , i

j (m+1) i k + (γ i jk ∘ 𝜘 t ) V M;X V M;X , (A M;X ) = V̇ M;X

i = 1, . . . , m .

2 The main aspects of Newtonian space-time are given in Chapter 2, Section 2.5. 3 We use upper index m + 1 to indicate that the corresponding mapping takes values in space-time.

6.4 Flows and Lie Derivatives

|

165

Finally, one obtains the sought acceleration of the point X: A M;X : 𝕋 → TP ,

i k A iM;X = V̇ M;X + γ i jk V M;X V M;X , j

i = 1, . . . , m .

The field A : B × 𝕋 → TP ,

A(X, t) := A M;X (t) ,

is called the material acceleration. The spatial acceleration is the map ̂ −1 a t = A(⋅, t) ∘ 𝜘 t : S𝜘t → TP . Since V(X, t) = v(χ M;X (t), t), one has V̇ i = v̇ i + V j ∂ j v i . This relation implies that a it = v̇ i + v j ∂ j v i + γ i jk v j v k . If dim B = dim P, then any shape is an open set. In this case, the latter expression can be represented in the component-free form at =

∂v t + ∇v t v t . ∂t

Here, ∂v t /∂t is calculated with respect to a fixed spatial point.

6.4 Flows and Lie Derivatives The preceding sections show the close relation between a motion and a continual set of curves, which represent spatial trajectories of material points. Such trajectories are integral curves for the material velocity field, and the latter is its infinitesimal generator. At the same time, the notion of flow is similarly based on the continual set of curves. The theory of flows on smooth manifolds gives a tool that is necessary for formulating the transport theorem and balance equations on smooth manifolds, which represent the body and the physical space. Such a tool is a Lie derivative. The main aspects of flows and Lie derivatives are briefly considered below.

6.4.1 Vector Fields and Integral Curves Suppose that M is a smooth s-dimensional manifold. A vector field on M is a section u : M → TM of the tangent bundle TM. The material and spatial velocities give the examples of mappings that have TM as a codomain, but their domains are contained in the Cartesian product M × ℝ. Thus, such mappings do not satisfy the definition of a

166 | 6 Motion vector field. To this end, we introduce the following definition. Suppose that U ⊂ M×ℝ is an open subset. A time-dependent vector field on M [3] is a continuous mapping u : U → TM, such that ∀(p, t) ∈ U : u(p, t) ∈ T p M . If t ∈ ℝ is such that the set Ut = {p ∈ M | (p, t) ∈ U} is not empty, then by definition the continuous mapping u t = u(⋅, t) : Ut → TM is a vector field in the conventional sense. Any smooth curve χ : 𝕀 → M generates the velocity vector field t 󳨃→ χ󸀠 (t). Vice versa, a vector field is given and one needs to obtain a curve for which such a field is the velocity vector field. Suppose that u : M → TM is a vector field. An integral curve of u [3] is a smooth curve χ : 𝕀 → M with the property ∀t ∈ 𝕀 :

χ󸀠 (t) = u χ(t) .

For the sake of simplicity, assume that 0 ∈ 𝕀. The point χ(0) is called the starting point of χ. The question of determining the integral curve χ, which has a point p ∈ M as its starting point, reduces to the solution of the system of ordinary differential equations (ODE) system with an initial condition. Choose a smooth chart (U, φ), such that p ∈ U. Then, in local coordinates one has φ ∘ χ(t) = (χ1 (t), . . . , χ s (t)) and u = u i ∂ i . One obtains the following autonomous ODE system: dχ i = u i (χ 1 , . . . , χ s ) , dt

i = 1, . . . , s ,

with the initial condition (χ1 (0), . . . , χ s (0)) = φ(p). The solution for this system is defined in an open interval ] − ε, ε[, for some ε > 0. The curve χ is the sought integral curve of u that starts at p. Let u : U → TM be a time-dependent vector field. An integral curve of u is a smooth curve χ : 𝕀 → M with the property ∀t ∈ 𝕀 : ((t, χ(t)) ∈ U) ∧ (χ 󸀠 (t) = u(χ(t), t)) . Choose a smooth chart (U, φ). Then, in local coordinates one has φ ∘ χ(t) = (χ 1 (t), . . . , χ s (t)) and u = u i ∂ i . One obtains the following non-autonomous ODE system: dχ i = u i (χ 1 , . . . , χ s , t) , dt

i = 1, . . . , s .

6.4.2 Flow A global flow on a smooth manifold M is a continuous left action of the additive group ℝ on M. In other words, it is a continuous map F : ℝ × M → M, which satisfies the properties ∀p ∈ M ∀t, s ∈ ℝ : F(t, F(s, p)) = F(t + s, p) , ∀p ∈ M :

F(0, p) = p .

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A global flow F : ℝ × M → M, (t, p) 󳨃→ F(t, p) defines two types of maps, whether t or p is fixed. For any t ∈ ℝ, one has the continuous mapping F t : M → M, such that F t = F(t, ⋅). It satisfies the group property F t ∘ F s = F t+s ,

F0 = IdM .

For each p ∈ M, one has the curve F (p) : ℝ → M, F (p) = F(⋅, p). For any p ∈ M, define u p = (F (p) )󸀠 (0) ∈ T p M. If F is a smooth global flow, then the assignment u : p 󳨃→ u p is a smooth vector field on M. Each curve F (p) is an integral curve of u [3]. Due to this fact, the vector field u is called the infinitesimal generator of F. Suppose that u ∈ Vec(M). The integral curve of u is defined from the ODE system with an initial condition. This initial value problem has the unique solution defined on some open interval, which does not coincide with ℝ, in general. This means that u cannot be an infinitesimal generator of a global flow F, since every curve F (p) must be defined on ℝ. For this reason, we introduce the notion of local flow [3]. Let M be a smooth manifold. A flow domain is an open subset G ⊂ ℝ × M, such that for all p ∈ M, the set G(p) = {t ∈ ℝ | (t, p) ∈ G} is an open interval containing⁴ 0. A local flow on M is a continuous mapping F : G → M that satisfies the following properties: for all p ∈ M one has F(0, p) = p , and for all s ∈ G(p) and t ∈ G(F(s,p)), such that s + t ∈ G(p) , F(t, F(s, p)) = F(t + s, p) . For all t ∈ ℝ, define the set Mt = {p ∈ M | (t, p) ∈ G} . As for global flow, define the mapping F t : Mt → M as F t (p) := F(t, p) and the curve F (p) : G(p) → M as F (p) (t) = F(t, p). If F is a smooth local flow, the infinitesimal generator of F is defined similarly as for global flow: u p = (F (p) )󸀠 (0) ∈ T p M. Then, one has that u : p 󳨃→ u p is a smooth vector field on M, and each curve F (p) is an integral curve of u [3]. On the contrary, suppose that we are given a vector field u ∈ Vec(M). There is a unique smooth local flow F : G → M whose infinitesimal generator is u. This flow has the following properties⁵: (i) F cannot be extended to a flow on a larger flow domain. (ii) For any p ∈ M, the curve F (p) : G(p) → M is the unique integral curve of u that starts at p. This curve cannot be extended to an integral curve on any larger open interval. 4 It follows that {0} × M ⊂ G. 5 These properties constitute the theorem that is proven in [3].

168 | 6 Motion (iii) For any p ∈ M, if s ∈ G(p) , then G(F(s,p)) = {t − s | t ∈ G(p) }. (iv) For each t ∈ ℝ, the set Mt is open in M. The mapping F t : Mt → M−t (the codomain of the original map was restricted to Mt ) is a diffeomorphism with inverse F−t .

6.4.3 Lie Derivatives Consider a pair of vector fields u, v ∈ Vec(M) on a smooth manifold M. The vector field u generates the local flow F : G → M. Let p ∈ M. The sets Mt , M−t contain p for values of t that belong to a sufficiently small interval ] − ε, ε[ for ε > 0. Since⁶ T p F t : T p M → T F t (p) M and T F t (p) F−t : T F t (p) M → T p M, one obtains that (T p F t )v p ∈ T F t (p) M and (T F t (p) F−t )v F t (p) ∈ T p M. The Lie derivative of v with respect to u at point p has two equivalent definitions [3, 122, 123]: (v ∘ F t )p − (T p F t )v p , t→0 t (T F t (p) F−t )v F t (p) − v p , (Lu v)p = lim t→0 t

(Lu v)p = lim

(6.1) (6.2)

and if these limits exist, (6.1) and (6.2) represent a vector from T p M. Formula (6.1) defines the Lie derivative by means of a pushforward, while (6.2) defines the Lie derivative as a pullback. The latter definition can be rewritten as follows⁷: (Lu v)p = lim

t→0

(F ∗t v) p − v p t

.

(6.3)

Here, (F ∗t v)p denotes the value of F ∗t v at point p. The mapping p 󳨃→ (Lu v)p is well defined and represents a smooth vector field on M [3]. This field is denoted by Lu v. Definitions (6.1) and (6.2) (or (6.3)) require finding local flows that are generated by u. Such an operation may be cumbersome in the computational aspect, but, since the Lie derivative can be defined as a limit at t → 0, its calculation can be carried out in a simpler way. In order to do this, one needs to take into account the principal parts of u, v only, when the expansion with respect to t is performed (see [122, p. 126]): Lu v = [u, v] , where [u, v] are Lie brackets (13.20). As an example, calculate Lie derivatives directly, by (6.1), (6.2), and with the definition of Lie brackets. Suppose that E is a two-dimensional affine Euclidean space

6 The sets Mt and Mt are open submanifolds of M. Since p ∈ Mt and p ∈ M−t , one has T p Mt = T p M−t = T p M. 7 Recall that F −1 = F−t . Thus, F ∗t v = TF−t ∘ v ∘ F t (for the definition of pullback, see Chapter 13, t Section 13.3.7).

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with a fixed Cartesian coordinate system (o, (i 1 , i2 )). Let O be a set O = E \ {x ∈ E | p(x) = x1 i1 , x1 ≥ 0} . It is open, because it was obtained from E by puncturing points from the axis ox1 that are in the positive direction. Induce the topological structure from E on O. Consider a smooth atlas on O, which consists of two charts. The first one is standard⁸, (O, D). The second one is (O, H), where H = h−1 ∘ D, and the mapping h : ]0, +∞[ × ]0, 2π[ → ℝ2 ,

(r, φ) 󳨃→ (x1 , x2 )

is defined by the rule x1 = r cos φ, x2 = r sin φ. Each such chart covers the whole O, and this means that one has the transformation of variables on O. The symbols ∂1 , ∂2 denote the coordinate frame that corresponds to the chart (O, D), and the symbols ∂ r , ∂ φ denote the coordinate frame in the j chart (O, H). Using the formula ∂ i󸀠 = ∂xi󸀠 ∂ j , which relates “old” frame (∂ j ) to the “new” ∂x

󸀠

󸀠

one, (∂ i󸀠 ), with respect to a transformation x i = x i (x j ), one arrives at the relations ∂ r = cos φ∂1 + sin φ∂2 ,

∂ φ = −r sin φ∂1 + r cos φ∂2 .

In addition, introduce the field e⟨φ⟩ = 1r ∂ φ . For a geometric interpretation, one needs to take into account the correspondences i1 ↔ ∂1 , i2 ↔ ∂2 . To this end, define the mappings Φ t : (r, φ) 󳨃→ (R, Ψ) , which are coordinate representations of flows in the chart (O, H), calculate Lie derivatives at point p = H −1 (r, φ) according to (6.1), (6.2), and compare them with similar derivations with Lie brackets. The map f t = H −1 ∘Φ t ∘H represents a flow that meets Φ t . (i) Define the flow that is generated by ∂ r . Its coordinate representation can be obtained from the ODE system dr =1, dt

dφ =0, dt

with initial conditions r = r0 , φ = φ0 . The solution for this system has the form Φ t (r0 , φ0 ) = (r0 + t, φ0 ) . Hence, one obtains the mapping (r, φ) 󳨃→ Φ t (r, φ), which has the following coordinate representation in (x1 , x2 ): (r, φ) 󳨃→ Ξ t (r, φ) = ((r + t) cos φ, (r + t) sin φ) .

8 Here the symbol D is used for the restriction of the mapping D(o,(i 1 ,i 2 )) (2.2) to O.

170 | 6 Motion Calculate the Lie derivatives L∂ r ∂ r , L∂ r ∂ φ and L∂ r e⟨φ⟩ at point p using formula (6.1). The tangent map Tf t : TO → TO has the form Tf t = ∂ R ⊗ dr + ∂ Ψ ⊗ dφ , where (R, Ψ) = (r + t, φ). Hence, Tf t ∂ r = ∂ R ,

Tf t ∂ φ = ∂ Ψ ,

Since ∂ r ∘ f t = ∂ R , ∂ φ ∘ f t = ∂ Ψ , e⟨φ⟩ ∘ f t =

1 R ∂Ψ ,

Tf t e⟨φ⟩ = 1r ∂ Ψ . one has

∂ R − Tf t ∂ r =0; t ∂ Ψ − Tf t ∂ φ = 0; [L∂ r ∂ φ ]p = lim t→0 t 1 ∂ Ψ − Tf t e⟨φ⟩ [L∂ r e⟨φ⟩ ]p = lim R = lim t→0 t→0 t [L∂ r ∂ r ]p = lim

t→0

1 r+t ∂ Ψ

− 1r ∂ Ψ 1 = − e⟨φ⟩ . t r

A similar result can be obtained if one calculates the Lie brackets: [∂ r , ∂ r ] = ∂ r ∂ r − ∂ r ∂ r = 0 , [∂ r , ∂ φ ] = ∂ r ∂ φ − ∂ φ ∂ r = 0 , 1 [∂ r , e⟨φ⟩ ] = − e⟨φ⟩ . r Now calculate the Lie derivatives via formula (6.2). The inverse to the coordinate representation of the flow Φ t has the form Φ−1 t (R, Ψ) = (R − t, Ψ) ,

R=r+t,

Ψ =φ,

at a point with polar coordinates (R, Ψ). The tangent map to f t−1 is Tf t−1 = ∂ r ⊗ dR + ∂ φ ⊗ dΨ . For the vector field ∂ φ , one has {∂ φ ∘ f t }p = ∂ Ψ , and hence, {(f t )∗ ∂ φ }q = {∂ r ⊗ dR + ∂ φ ⊗ dΨ}∂ Ψ = ∂ φ . Thus, the Lie derivative is determined by the expressions [L∂ r ∂ φ ]p = lim

{(f t )∗ ∂ φ }p − ∂ φ t

t→0

For the vector field e⟨φ⟩ , one has {e⟨φ⟩ ∘ f t }p =

= lim

t→0

1 R ∂ Ψ . Its

∂φ − ∂φ =0. t pullback is equal to

{(f t )∗ e⟨φ⟩ } p = {∂ r ⊗ dR + ∂ φ ⊗ dΨ} R1 ∂ Ψ =

r e⟨φ⟩ , r+t

and, hence, the Lie derivative is equal to [L∂ r e⟨φ⟩ ]p = lim

t→0

{(f t )∗ e⟨φ⟩ } p − e⟨φ⟩ t

= − lim

t→0

1 1 e⟨φ⟩ = − e⟨φ⟩ . r+t r

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(ii) For the flow generated by ∂ φ , one has the ODE system dr =0, dt

dφ =1, dt

with initial conditions r = r0 , φ = φ0 . The solution for this system gives the coordinate representation for the flow: Φ t (r0 , φ0 ) = (r0 , t + φ0 ) . In coordinates (x1 , x2 ) one has another representation: (r, φ) 󳨃→ Ξ t (r, φ) = (r cos(t + φ), r sin(t + φ)) . Calculate L∂ φ ∂ r at point p according to formula (6.1). The tangent map Tf t : TO → TO has the form Tf t = ∂ R ⊗ dr + ∂ Ψ ⊗ dφ , where (R, Ψ) = (r, t + φ). Thus, one obtains Tf t ∂ r = ∂ R . Since {∂ r ∘ f t }p = ∂ R , one arrives at the equalities [L∂ φ ∂ r ]p = lim

t→0

∂ R − Tf t ∂ r =0. t

Calculation of the Lie brackets gives a similar result: [∂ φ , ∂ r ] = ∂ φ ∂ r − ∂ φ ∂ r = 0 . Now, use the formula (6.2) to calculate the Lie derivatives. The mapping, which is inverse to the coordinate representation of the flow Φ t , has the following expression at point with polar coordinates (R, Ψ): Φ−1 t (t) = (R, Ψ − t) ,

r=R,

Ψ =φ+t.

The corresponding tangent map is represented by the expression Tf t−1 = ∂ r ⊗ dR + ∂ φ ⊗ dΨ , which implies the expression for values of pullback for ∂ r at point p: {f t∗ ∂ r }p = {∂ r ⊗ dR + ∂ φ ⊗ dΨ}∂ R = ∂ r , and the expression for the Lie derivative of ∂ r among ∂ φ : [L∂ φ ∂ r ]p = lim

t→0

{f t∗ ∂ r } p − ∂ r t

= lim

t→0

∂r − ∂r =0. t

172 | 6 Motion

(iii) Finally, for the coordinate representation of the flow generated by e⟨φ⟩ , we have the ODE system dφ 1 dr =0, = , dt dt r with the initial conditions r = r0 , φ = φ0 . Its solution has the form Φ t (r0 , φ0 ) = (r0 ,

t r0

+ φ0 ) .

Correspondingly, the flow, which is generated by e⟨φ⟩ , can be represented by the expression (r, φ) 󳨃→ Φ t (r, φ). Calculate the Lie derivative for ∂ r with respect to e⟨φ⟩ , i.e., Le⟨φ⟩ ∂ r , at point p using (6.1). One has the following expression for the tangent map Tf t : TO → TO: Tf t = ∂ R ⊗ dr −

t ∂ Ψ ⊗ dr + ∂ Ψ ⊗ dφ , r2

where (R, Ψ) = (r, rt + φ). Thus, one obtains Tf t ∂ r = ∂ R −

t ∂Ψ . r2

Since {∂ r ∘ f t }p = ∂ R , and due to the continuity of ∂ φ , one has [Le⟨φ⟩ ∂ r ]p = lim

t→0

∂ R − Tf t ∂ r 1 1 = 2 lim ∂ t +φ = e⟨φ⟩ . t r r t→0 r

As was expected, the Lie brackets from the vector fields e⟨φ⟩ and ∂ r give the same result: 1 1 1 [e⟨φ⟩ , ∂ r ] = ∂ φ ∂ r − ∂ r ( ∂ φ ) = e⟨φ⟩ . r r r Let us do the final calculation of the Lie derivative using (6.2). The inverse mapping to the coordinate representation of the flow Φ t at point with polar coordinates (R, Ψ) has the form Φ−1 t (R, Ψ) = (R, −

t + Ψ) , R

R=r,

Ψ =φ+t.

Taking this into account, one obtains the expression for the tangent map: Tf t−1 = ∂ r ⊗ dR +

t ∂ φ ⊗ dR + ∂ φ ⊗ dΨ . R2

The relations obtained allow one to represent pullback of the field ∂ r in the form {f t∗ ∂ r }p = ∂ r + hence, [Le⟨φ⟩ ∂ r ]p = lim

t→0

{f t∗ ∂ r } p − ∂ r t

t ∂φ , (r + t)2

= lim

t→0

r 1 e⟨φ⟩ = e⟨φ⟩ . r (r + t)2

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173

(x, y) → (r, ϕ)

(x, y) → (i1 , i2 )

(r, ϕ) → (er , eϕ ) (ei ) → (ei )

eϕ

ft

e

ft ϕ ◦ fter

er ft

◦

e r

fter

eϕ ft



e

ft

f

t

ft ϕ

e

r



◦f

t

e

ϕ



e

e

ft ϕ ◦ ft r

Fig. 6.1: Flows and Lie derivatives.

The geometric interpretation of the relations obtained is shown in Figure 6.1. It contains the spaceE, with two coordinate systems in its part: the Cartesian (x, y), with the basis (i 1 , i 2 ), and a polar one, (r, φ), with the frame (e r , e φ ), which is the result of embedding (∂ r , ∂ φ ) into E. Coordinate grids of both systems are shown: rectangular and a system of concentric circles and rays. The pair (e ⟨r⟩ , e⟨φ⟩ ) is the frame, obtained from the corresponding polar frame by normalizing its vectors. The lower half of the figure is devoted to the construction of flows along the corresponding vector fields. In the case when the flows were considered along the basis vectors of the polar system, a closed parallelogram was constructed. A similar case for a normalized frame leads to an unclosed parallelogram. In the second case, the Lie derivative is different from zero. This illustrates the argument that the Lie derivative is a measure of the non-closure of a parallelogram constructed on curves generated by flows. The Lie derivative is generalized to arbitrary tensor fields by a formula that is analogous to (6.3). Suppose that u ∈ Vec(M) and F : G → M is the local flow generated by u. Let T ∈ Sec(T (k,l) (TM)). Then, the Lie derivative of T with respect to u at a point

174 | 6 Motion p ∈ M is the limit [3, 123] (Lu T)p = lim

(F ∗t T)p − Tp

t→0

t

.

In the framework of calculus on the manifolds, it is shown that (Lu T)p exists for all p ∈ M, and the assignment p 󳨃→ (Lu T)p is an element of Sec(T (k,l) (TM)). Such an assignment is denoted by Lu T. The operation Lu satisfies the following properties [3]: (i) ∀f ∈ C∞ (M) : Lu f = u(f); (ii) ∀f ∈ C ∞ (M) ∀T ∈ Sec(T (k,l) (TM)) : Lu (f T) = (Lu f)T + f Lu T; (iii) ∀T, Q ∈ Sec(T (k,l) (TM)) : Lu (T ⊗ Q) = (Lu T) ⊗ Q + T ⊗ Lu Q; (iv) ∀f ∈ C∞ (M) : Lu (df) = d(Lu f). In particular, these properties imply that if g = g ij dx i ⊗ dx j is a Riemannian metric on M, then its Lie derivative is represented by the expression Lu g = {u k ∂ k g ij + g kj ∂ i u k + g ik ∂ j u k } dx i ⊗ dx j .

(6.4)

Consider the case of differential forms, which are the elements of Ω k (M). The Lie derivative is defined in the similar way as for arbitrary smooth tensor fields. Suppose that u ∈ Vec(M). The operation Lu has the following properties [3]: (i) ∀ω ∈ Ω k (M) ∀η ∈ Ω l (M) : Lu (ω ∧ η) = (Lu ω) ∧ η + ω ∧ (Lu η); (ii) for all ω ∈ Ω k (M) Cartan’s magic formula holds: Lu ω = i u (dω) + d(i u ω) .

(6.5)

6.4.4 Time-Dependent Flow A flow, in some sense, is a family of integral curves of a vector field on a smooth manifold. Thus, a family of autonomous ODE systems corresponds to a flow. A timedependent flow is similarly a family of integral curves, but for time-dependent vector fields. In this case, one deals with a family of non-autonomous ODE systems. Let 𝕀 ⊂ ℝ be an open interval and G ⊂ 𝕀 × 𝕀 × M an open subset. Suppose that for each t0 ∈ 𝕀 and p ∈ M the set G(t 0 ,p) = {t ∈ 𝕀 | (t, t0 , p) ∈ G} is an open interval that contains⁹ t0 . Also define the set Mt 0 ,t 1 = {p ∈ M | (t0 , t1 , p) ∈ G} ,

9 In particular, this implies that {t0 } × {t0 } × M ⊂ G.

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175

for each t0 , t1 ∈ 𝕀. A time-dependent flow [3, 123] is a continuous mapping F : G → M with the properties: for all p ∈ M and t0 ∈ 𝕀, one has F(t0 , t0 , p) = p , and if p ∈ Mt 1 ,t 0 and F(t1 , t0 , p) ∈ Mt 2 ,t 1 , then F(t2 , t1 , F(t1 , t0 , p)) = F(t2 , t0 , p) . For each t0 , t1 ∈ 𝕀, let us introduce the map F t 0 ,t 1 : Mt 0 ,t 1 → M by the relation F t 0 ,t 1 (p) := F(t0 , t1 , p). Then, the properties of F take the form F t 2 ,t 1 ∘ F t 1 ,t 0 (p) = F t 2 ,t 0 (p) ,

F t 0 ,t 0 (p) = p .

Thus, a time-dependent flow can be considered as a two-parametric family {F t 0,t 1 }t 0 ,t 1∈𝕀 of mappings F t 0 ,t 1 . This family is called a time-dependent flow itself in [123]. A flow F generates for every t0 ∈ 𝕀 and p ∈ M a curve F (t 0 ,p) : G(t 0 ,p) → M by the rule F (t 0 ,p)(t) := F(t, t0 , p). One has F (t 0 ,p)(t0 ) = p. If F is smooth, then one can consider the velocity vector u p,t 0 = (F (t 0 ,p) )󸀠 (t0 ). The assignment (p, t) 󳨃→ u p,t is a time-dependent vector field on M × 𝕀. Conversely, suppose that u : M × 𝕀 → M is a smooth time-dependent vector field. There exists an open subset G ⊂ 𝕀 × 𝕀 × M and a smooth map F : G → M that satisfies the following properties¹⁰: (i) For any t0 ∈ 𝕀 and p ∈ M the set G(t 0 ,p) is open interval that contains t0 . The curve F (t 0 ,p) : G(t 0 ,p) → M is smooth and represents the unique integral curve of u with the initial condition F (t 0 ,p) (t0 ) = p. This curve cannot be extended to an integral curve on any larger open interval. (ii) If t1 ∈ G(t 0 ,p) and q = F (t 0 ,p) (t1 ), then G(t 1 ,q) = G(t 0 ,p) ,

F (t 1 ,q) = F (t 0 ,p) .

(iii) The set Mt 0 ,t 1 is an open subset in M for all t0 , t1 ∈ 𝕀. The mapping F t 0 ,t 1 : Mt 0 ,t 1 → M is a diffeomorphism from Mt 0 ,t 1 onto Mt 1 ,t 0 . Its inverse is F t 1 ,t 0 . (iv) If p ∈ Mt 1 ,t 0 and F(t1 , t0 , p) ∈ Mt 2 ,t 1 then p ∈ Mt 2 ,t 0 and F t 2 ,t 1 ∘ F t 1 ,t 0 (p) = F t 2 ,t 0 (p) . This map, F, is the time-dependent flow of u. Let F be a time-dependent flow, which is generated by a smooth time-dependent vector field u : M × 𝕀 → TM, and Tt a smooth tensor field, which depends on points

10 These properties constitute the theorem proved in [3].

176 | 6 Motion from M and the numerical parameter t ∈ 𝕀. The autonomous Lie derivative [60] is defined via relation 󵄨󵄨 d ∗ 󵄨 {F t;s Ts }󵄨󵄨󵄨 , (6.6) Lu Tt = 󵄨󵄨t=s dt and the consequence of such a definition is the equality [123]: d ∗ ∂Tt F Tt = F ∗t,s Lu Tt + F ∗t,s , dt t,s ∂t

(6.7)

which is used in this book. In the equality (6.7), one needs clarification for the sym∂u t bol ∂T ∂t . We give such a clarification for the case ∂t , where u : M×𝕀 → TM; the general case is considered in an analogous way. In calculating the partial derivative of some function, one means that other variables of such a function are fixed. In our considerations, the variable p is fixed. This means that the mapping u(p, ⋅) : 𝕀 → TM takes values on the same vector space, T p M, and the expression u(p,t+h)−u(p,t) makes sense. h Choose some Riemannian metric g on M. At a point p it represents a scalar product. The convergence of the function u(p,t+h)−u(p,t) to some vector from T p M as h → 0 is unh derstood as the norm convergence, induced by scalar product. The symbol ∂u ∂t denotes i

s such a vector. Its coordinate representation has the form ( ∂u ∂t )i=1 . The Lie derivative from a time-dependent tensor field T is the operation 𝕃 that is defined by the formula [60]: 󵄨󵄨󵄨 d 𝕃 u Tt = . (6.8) {F ∗t;s Tt }󵄨󵄨󵄨 󵄨󵄨t=s dt

The autonomous Lie derivative is related with 𝕃 as follows [60]: 𝕃 u Tt =

∂Tt + Lu Tt . ∂t

(6.9)

6.5 Motion as a Time-Dependent Flow Suppose that the body and the physical space have the same dimensions. Consider a smooth motion M = {𝜘 t }t∈𝕋 . It generates the family {̂ 𝜘t }t∈𝕋 of diffeomorphisms. Such a family, in turn, defines a two-parametric family {ψ t,s }t,s∈𝕋 of diffeomorphisms ̂t ∘ 𝜘 ̂ −1 ψ t,s = 𝜘 s : S𝜘s → S𝜘t , which satisfies the following relations: ∀t, s, q ∈ 𝕋 : ∀t ∈ 𝕋 :

ψ t,s ∘ ψ s,q = ψ t,q , ψ t,t = IdS𝜘t .

Thus, {ψ t,s }t,s∈𝕋 represents a time-dependent flow. Recall that the spatial velocity field is the mapping v : ⋃ (S𝜘t × {t}) → TP . t∈𝕋

6.5 Motion as a Time-Dependent Flow

| 177

̂ −1 Fix s ∈ 𝕋 and x ∈ S𝜘s . Denote X = 𝜘 s (x). Consider the smooth curve γ s;x : 𝕋 → P ,

γ s;x : t 󳨃→ ψ t,s (x) .

Such a curve coincides with the curve χ M;X . Thus, one obtains γ󸀠s;x (t) = V(X, t) for all t ∈ 𝕋, or, ∀t ∈ 𝕋 : γ 󸀠s;x (t) = v(γ s;x (t), t) , with the initial condition γ s;x (s) = x. Thus, γ s;x is an integral curve of v.

7 Stress Measures The conventional representations of stresses (Chapter 3) substantially use the Euclidean-affine structure of physical space, which contains both the reference and the actual shapes of the body. Below we list some characteristic features of this approach. ∙ The existence of the Cauchy stress tensor T is stated via the classical “Cauchy tetrahedron argument” that is based on the Euclidean structure. ∙ Integrals that represent the resulting contact forces essentially use the Euclidean parallel translation rule. Since contact force densities take values in V, one can sum these values taken at different points. Such summation is required by the definition of the integration in Euclidean space. ∙ Let S denote ∂𝜘 R (P) or ∂𝜘(P), where P ∈ Part(B). For a point x ∈ S, one has the decomposition T x S ⊕ (T x S)⊥ = V. Such a direct sum involves the Euclidean structure of V. Thus, n x denotes the unit normal vector, the element of (T x S)⊥ . In this chapter, the notion of stresses is generalized on smooth manifolds. We follow the ideas set out in [124].

7.1 Concentrated Forces and Force Densities Concentrated force. Let M = {𝜘t }t∈𝕋 be a smooth motion and X ∈ B be a material point. A concentrated force, which acts on the material point X at instant t ∈ 𝕋, is a covector f t ∈ T𝜘∗t (X) P. Suppose that a family {f t }t∈𝕋 of concentrated forces, which act on the material point X, corresponds to the given motion M. The motion M, in turn, generates the field {V M;t }t∈𝕋 of velocities V M;t ∈ Sec(𝜘∗t TP). For each t ∈ 𝕋, the scalar f t ⌞ V M;t (X) can be interpreted as the power that the force f expends at instant t. Finally, for each instant t ∈ 𝕋, a concentrated force that acts on some material point at that instant is represented by an element of the pullback bundle 𝜘∗t TP. Force density. As is usual in continuum mechanics, we assume that forces that act on the body B are divided into two groups. The first one is represented by body forces, which are distributed over the volume of any part of the body. The second group is represented by surface forces, which are distributed over the boundary of any part. Thus, both of these groups consist of force densities, which act on the velocity to produce power densities (volume or surface). The latter are represented by ordinary differential forms and can be integrated (over the part of the body or its boundary) to obtain the resulting powers. Suppose that M = {𝜘 t }t∈𝕋 is a smooth motion. Let P ∈ Part(B) be a part of the body B. Recall that by a part we mean an n-dimensional embedded submanifold with boundary. Consider force densities in material and spatial descriptions.

https://doi.org/10.1515/9783110563214-007

180 | 7 Stress Measures A body force density in material description is given by a family {β mat P;t } t∈𝕋 of sections ∗ ∗ n ∗ ∀t ∈ 𝕋 : β mat P;t ∈ Sec (𝜘 t T P ⊗ Λ (T P)) . A surface force density in material description is a family {τmat P;t } t∈𝕋 of sections ∗ ∗ n−1 ∗ τ mat (T ∂P)) . P;t ∈ Sec (𝜘 t T P ⊗ Λ

∀t ∈ 𝕋 :

We also postulate that in the material description, the resulting power expended by body and surface forces at instant t ∈ 𝕋 is of the form mat pmat := ∫ V M;t ⌟ β mat t P;t + ∫ V M;t ⌟ τ P;t . ∂P

P

spat

A body force density in spatial description is a family {β P;t }t∈𝕋 of sections β P;t ∈ Sec (ι∗𝜘t (P) T ∗ P ⊗ Λ n (T ∗ 𝜘 t (P))) . spat

∀t ∈ 𝕋 :

spat

A surface force density in spatial description is a family {τP;t }t∈𝕋 of sections τ P;t ∈ Sec (ι∗𝜘t (P) T ∗ P ⊗ Λ n−1 (T ∗ ∂𝜘 t (P))) . spat

∀t ∈ 𝕋 :

We postulate that in the spatial description, the resulting power expended by body and surface forces at instant t ∈ 𝕋 is of the form spat

pt

spat

:= ∫ v t ⌟ β P;t + 𝜘t (P)

∫

spat

v t ⌟ τP;t .

∂𝜘t (P)

Usually, it is assumed that the value of a body force at a point is independent from any part that contains this point: ∀t ∈ 𝕋 :

∈ Sec (𝜘∗t T ∗ P ⊗ Λ n (T ∗ B)) . β mat t

∀t ∈ 𝕋 :

βt

spat

∗ ∈ Sec (ιS T ∗ P ⊗ Λ n (T ∗ S𝜘t )) . 𝜘 t

In this regard, factually, only a surface force depends on a part. The collection {(β mat t , τmat )} represents a force system on B in material description, and the colP∈Part(B),t∈𝕋 P;t spat spat lection {(β t , τP;t )}P∈Part(B),t∈𝕋 represents a force system on B in spatial description. Thus, the total force F t , which acts on the part P at instant t ∈ 𝕋, is a functional over velocities. In material and spatial descriptions, it is represented by the values mat + ∫ V M;t ⌟ τmat F mat t (V) := ∫ V M;t ⌟ β t P;t ,

spat F t (v)

:= ∫ 𝜘t (P)

(7.1)

∂P

P spat vt ⌟ βt

+

∫ ∂𝜘t (P)

spat

v t ⌟ τ P;t .

(7.2)

7.2 Inclined Hyperplanes | 181

7.2 Inclined Hyperplanes The notion of the stress tensor presupposes the possibility to choose a hyperplane (surface element) with orientation. For this purpose, in Riemannian-oriented manifolds one can use a unit normal vector field. At the same time, the Riemannian metric G on the body B is obtained from certain physical reasonings. These reasonings may refer to the concept of stresses. In this regard, one needs a way to construct stresses on B without appealing to the metric. Particularly it means that the notion of the oriented hyperplane needs to be introduced in the metric-independent way. The required possible definition of the oriented hyperplane is based on the notion of the inclined hyperplane [124]. Let X ∈ B. Consider the tangent space TX B. By a hyperplane HX at the point X we mean an (n − 1)-dimensional vector subspace of TX B. Denote the collection of all hyperplanes at X by G n−1 (TX B). The set G n−1 (TX B) can be endowed by a smooth (n − 1)-dimensional manifold structure [3, 125]. This manifold is called the Grassmann manifold. The union G n−1 (TB) = ⨆ G n−1 (TX B) , X∈B

is the Grassmann bundle of hyperplanes. Remark 7.1. Let us briefly describe charts on G n−1 (TX B) [3]. For any one-dimensional subspace I ⊂ TX B, one can find a hyperplane HX such that TX B = HX ⊕ I. Let L ∈ Lin(HX ; I) be a linear map. Its graph, the set ΓL = {(u, L(u)) ∈ HX × I | u ∈ HX } , can be identified with the following (n − 1)-dimensional subspace of TX B: Φ(L) = {u + L(u) ∈ HX ⊕ I | u ∈ HX } . Such an identification is provided due to the fact that TX B can be considered as a result for both external and internal direct sums. One has that Φ(L) ∩ I = {0}. Conversely, let ∆ ⊂ TX B be an (n − 1)-dimensional subspace such that ∆ ∩ I = {0}. Denote by pr1 : TX B → HX and pr2 : TX B → I the projections that are determined by external direct sum decomposition. The map pr1 |∆ : ∆ → HX is injective by virtue of the condition ∆ ∩ I = {0}. Thus, pr1 |∆ is a vector space isomorphism. The mapping L := pr2 |∆ ∘ (pr1 |∆ )−1 : HX → I is linear and Φ(L) = ∆. Let U I be a subset of G n−1 (TX B), which consists of all (n − 1)-dimensional subspaces whose intersections with I consists of the zero vector only. The map Φ : Lin(HX ; I) → U I ,

u 󳨃→ Φ(L) ,

is a bijection. Denote φ := Φ−1 : U I → Lin(HX ; I). The space Lin(HX ; I) can be identified with ℝn−1 by choosing bases for HX and I. Thus, the ordered pair (U I , φ), where φ : U I → ℝn−1 , can be considered as a chart on G n−1 (TX B).

182 | 7 Stress Measures

Let HX be a hyperplane. Denote ∗ B | ∀u ∈ HX : ν(u) = 0} . Ann(HX ) := {ν ∈ TX

This is the vector space of all annihilators of HX . Since dim Ann(HX ) = 1, one has {TX B, ∀ν ∈ Ann(HX ) : ker ν = { H , { X

ν=0, ν ≠ 0 .

Using the space Ann(HX ) of annihilators one can introduce the notion of a side of HX . Define the following equivalence relation ∼HX on TX B \ HX : ∀u 1 , u 2 ∈ TX B \ HX :

(u 1 ∼HX u 2 ) ⇔ (∃ν ∈ Ann(HX ) :

ν(u 1 )ν(u 2 ) > 0) .

If ν(u 1 )ν(u 2 ) > 0 for some ν ∈ Ann(HX ) then it follows ν ≠ 0. Any other non-zero ̃ν ∈ Ann(HX ) can be expressed through ν: ̃ν = αν, for some α ∈ ℝ. Thus, ̃ν(u 1 )̃ν (u 2 ) > 0. The relation ∼HX is then well defined. If u 1 ∼HX u 2 , then we say that vectors u 1 and u 2 are on the same side of HX . If u 1 ≁HX u 2 , then we say that vectors u 1 and u 2 are on the opposite sides of HX . Note that if u ∈ TX B \ HX , then vectors u and −u are on the opposite sides. Thus, the quotient set consists of two classes – two sides of HX : (TX B \ HX )/ ∼HX = {[u], [−u]} . An inclined hyperplane is any of the pairs hX = (HX , [u]) and −hX = (HX , [−u]). The collection of all inclined hyperplanes at X is denoted by G⊥n−1 (TX B). One has the following natural map: pX : G⊥n−1 (TX B) → G n−1 (TX B) ,

pX (HX , [u]) := HX ,

which ignores the inclination. Again, let HX be a hyperplane. Choosing a positive inclination means choosing exactly one of classes from (TX B \ HX )/ ∼HX . A vector u is positively inclined if it belongs to the equivalence class that corresponds to the positive inclination. Let hX = (HX , [v]) be the hyperplane HX , together with chosen positive inclination. Define the set Ann+ (HX ) := {ν ∈ Ann(HX ) | ∀u ∈ [v] : ν(u) > 0} . ∗ B \ {0}: Introduce the equivalence relation ∼X,+ on TX ∗ B \ {0} : (ν1 ∼X,+ ν2 ) ⇔ (∃a > 0 : ν2 = aν1 ) . ∀ν1 , ν2 ∈ TX

By the definition, each pair of elements from Ann+ (HX ) is ∼X,+ -related. This implies ∗ ∗ B \ {0})/ ∼X,+ . Since any non-zero covector ν ∈ TX B defines the Ann+ (HX ) ∈ (TX ⊥ ∗ unique hyperplane, ker ν, one can identify spaces G n−1 (TX B) and (TX B \ {0})/ ∼X,+ .

7.3 Piola Section on Inclined Hyperplanes | 183

Choosing any inner product¹ on TX B, one can identify Ann+ (HX ) with a unit normal vector to HX . Thus, G⊥n−1 (TX B) can be given smooth structure of (n−1)-sphere. Denote G⊥n−1 (TB) = ⨆ G⊥n−1 (TX B) . X∈B

One has the bundle π : G⊥n−1 (TB) → B and the bundle morphism p : G⊥n−1 (TB) → G n−1 (TB) . Let P ∈ Part(B). Choose a point X ∈ ∂P. Consider the set CuX;P of all possible smooth curves χ : [0, ε[→ B, ε > 0 that satisfy the conditions: (i) χ(0) = X; (ii) χ(]0, ε[) ⊂ B \ P. In other words, all curves from the set CuX;P pass through X and lie outside P. Each vector from² TX B \ TX ∂P that is tangent to some curve χ from CuX;P at X, can be declared outward pointing (relatively to P). If u is an outward pointing vector, then choose the inclined hyperplane hX := (TX ∂P, [u]). Repeating this procedure for all X ∈ ∂P results in the desired collection {hX }X∈∂P of “material outwardly oriented area elements”.

7.3 Piola Section on Inclined Hyperplanes Again, we suppose that no Riemannian metric on the body B is defined. Let M = {𝜘 t }t∈𝕋 be a smooth motion. The introduced notion of the inclined hyperplane allows one to give an exact meaning for the Cauchy postulate. Interaction density bundle. Consider the vector bundle π : 𝜘∗t T ∗ P ⊗ Λ n−1 (G∗n−1 (TB)) → G n−1 (TB) , ∗ whose fiber is T𝜘∗t (X) P ⊗ Λ n−1 (HX ). The pullback of this bundle by means of the mor⊥ phism p : G n−1 (TB) → G n−1 (TB) gives

̃ : p∗ (𝜘∗t T ∗ P ⊗ Λ n−1 (G∗n−1 (TB))) → G⊥n−1 (TB) . π Following [124], we refer to this bundle as the interaction density bundle. All is ready for the mathematical formalization of the Cauchy postulate.

1 This inner product plays a technical role and does not require the metric on B. 2 Here we consider T X ∂P as an (n − 1)-dimensional hyperplane of T X B.

184 | 7 Stress Measures The Cauchy postulate. For each instant t ∈ 𝕋, there exists a section τ Pt : G⊥n−1 (TB) → p∗ (𝜘∗t T ∗ P ⊗ Λ n−1 (G∗n−1 (TB))) , called the Piola section³, such that for each part P ∈ Part(B) and the corresponding family {hX }X∈∂P of inclined hyperplanes for ∂P, one has ∀X ∈ ∂P :

P τ mat P;t (X) = τ t (h X ) .

The Piola stress form. Our next purpose is to relate τmat P;t , t ∈ 𝕋, with a covector-valued Piola stress form P t ∈ Sec (𝜘∗t T ∗ P ⊗ Λ n−1 (T ∗ B)) , (7.3) defined on B. To this end, one can use the notion of inclined restriction [124]. For any X ∈ B, let hX = (HX , [v]) be some inclined hyperplane. The inclusion mapping ι HX : HX → TX B generates the mapping ∗ ∗ B) → T𝜘∗t (X) P ⊗ Λ n−1 (HX ) , I ∗ : T𝜘∗t (X) P ⊗ Λ n−1 (TX

such that I ∗ (ω)(u 1 , . . . , u n−1 ) := ω(ι HX u 1 , . . . , ι HX u n−1 ) . The inclined restriction of P t (X), the element ∗ ι∗hX (P t (X)) ∈ T𝜘∗t (X) P ⊗ Λ n−1 (HX ) ,

is defined as follows: ι∗hX (P t (X))(u 1 , . . . , u n−1 ) := I ∗ (P t (X))(u 1 , . . . , u n−1 ) , if (v, u 1 , . . . , u n−1 ) are positively oriented, and ι∗hX (P t (X))(u 1 , . . . , u n−1 ) := −I ∗ (P t (X))(u 1 , . . . , u n−1 ) , if (v, u 1 , . . . , u n−1 ) are negatively oriented. This definition implies the Newton’s III law ι∗−hX (P t (X)) = −ι∗hX (P t (X)) . The layerwise mapping h 󳨃→ ι∗h (P t ) plays the role of a Piola section. Let P ∈ Part(B) and let {hX }X∈∂P be a collection of the corresponding inclined hyperplanes. One arrives at the desired relation [124] ∗ τ mat P;t = ι h (P t ) ,

or, pointwise, ∀X ∈ ∂P :

∗ τmat P;t (X) = ι h X (P t (X))

This is the generalization of the relation τ mat P;t = P t N in Euclidean mechanics.

3 In [124], the section τ t is called the Cauchy section. Since we are dealing with a material description, the term “Piola section” is more preferable.

7.4 The Cauchy Section

| 185

7.4 The Cauchy Section One needs to consider shapes of the body B and its parts with the aim of introducing the notion of the Cauchy stress form. In contrast to the case of the Piola stress form, one can use the additional data given by the Riemannian metric g of the physical space P. In this book, we use the approach of inclined hyperplanes for both the material and the spatial description. The reduction for the metric case can be found in [124]. Inclined hyperplanes for shapes. Let M = {𝜘 t }t∈𝕋 be a smooth motion. Fix t ∈ 𝕋 and consider the shape S𝜘t of the body B. If x ∈ S𝜘t , then the collection of all (n − 1)dimensional vector subspaces H x ⊂ T x S𝜘t (hyperplanes at x) form the Grassmann manifold G n−1 (T x S𝜘t ). Thus, the Grassmann bundle of hyperplanes is G n−1 (S𝜘t ) = ⨆ x∈S𝜘 G n−1 (T x S𝜘t ). t Since the physical space P is equipped with metric g, one can pull back this met∗ ric on S𝜘t : g(S𝜘t ) = ιS g. Let x ∈ S𝜘t . For any hyperplane H x ⊂ T x S𝜘t one can write 𝜘t ⊥ the decomposition T x S𝜘t = H x ⊕ H ⊥ x , where H x is the one-dimensional vector space, such that (S𝜘t ) H⊥ (v, u) = 0} . x := {u ∈ T x S𝜘t | ∀v ∈ H x : g (S𝜘 )

t ♭ (n, ⋅) annihiChoose some vector n ∈ H ⊥ x . By the definition, the covector n := gx ♭ ♭ lates H x ; ker n = H x . This implies n ∈ Ann(HX ). Thus, the covector n♭ may be used to test vectors u 1 , u 2 ∈ T x S𝜘t \ H x for being on the same side of H x (n♭ (u 1 )n♭ (u 2 ) > 0) or being on the opposite sides of H x (n♭ (u 1 )n♭ (u 2 ) < 0). The equivalence relation ∼H x on T x S𝜘t \ H x can be introduced in a specific way:

∀u 1 , u 2 ∈ T x S𝜘t \ H x :

(u 1 ∼H x u 2 ) ⇔ (n♭ (u 1 )n♭ (u 2 ) > 0) .

The quotient set consists of two classes – two sides of H x : (T x S𝜘t \ H x )/ ∼H x = {[u], [−u]} . An inclined hyperplane is any of the pairs h x = (H x , [u]) and −h x = (H x , [−u]). One arrives at the collection G⊥n−1 (T x S𝜘t ) of all inclined hyperplanes at x ∈ S𝜘t and defines the bundle G⊥n−1 (TS𝜘t ) = ⨆ G⊥n−1 (T x S𝜘t ) . x∈S𝜘t

The interaction density bundle and the boundary inclination. We have the vector bundle ∗ π : ιS T ∗ P ⊗ Λ n−1 (G∗n−1 (TS𝜘t )) → G n−1 (TS𝜘t ) , 𝜘 t

whose fiber is

T ∗x P

⊗

Λ n−1 (H ∗x ).

Introducing the morphism

p : G⊥n−1 (TS𝜘t ) → G n−1 (TS𝜘t ) ,

186 | 7 Stress Measures which layerwisely sends (H x , [u]) to H x , one arrives at the interaction density bundle ∗ ̃ : p∗ (ιS T ∗ P ⊗ Λ n−1 (G∗n−1 (TS𝜘t ) → G⊥n−1 (TS𝜘t ) . π 𝜘 t

Let t ∈ 𝕋 and P ∈ Part(B). Choose a point x ∈ ∂𝜘 t (P). As in material case, consider the set Cu x;𝜘t (P) of all possible smooth curves χ : [0, ε[→ S𝜘t , ε > 0, which satisfy the conditions: (i) χ(0) = x; (ii) χ(]0, ε[) ⊂ S𝜘t \ 𝜘 t (P). Each vector from T x S𝜘t \ T x ∂𝜘 t (P) that is tangent to some curve χ from Cu x;𝜘t (P) at x, can be declared outward pointing (relatively to 𝜘 t (P)). If u is an outward pointing vector, then choose the inclined hyperplane h t,x := (T x ∂𝜘 t (P), [u]). Repeating this procedure for all x ∈ ∂𝜘t (P) results in the desired collection {h t,x }x∈∂𝜘t (P) of “spatial outwardly-oriented area elements”. The Cauchy postulate. For each instant t ∈ 𝕋, there exists a section ∗ T ∗ P ⊗ Λ n−1 (G∗n−1 (TS𝜘t ))) , τ Ct : G⊥n−1 (TS𝜘t ) → p∗ (ιS 𝜘 t

called the Cauchy section, such that for each part P ∈ Part(B) and the corresponding family {h t,x }x∈S𝜘t of inclined hyperplanes built for boundary ∂𝜘t (P), one has ∀x ∈ ∂𝜘t (P) :

spat

τP;t (x) = τ Ct (h t,x ) .

The Cauchy stress form. In a similar manner to the material description, one can respat late τP;t , t ∈ 𝕋, with a covector-valued Cauchy stress form ∗ T t ∈ Sec (ιS T ∗ P ⊗ Λ n−1 (T ∗ S𝜘t )) , 𝜘 t

defined on S𝜘t . Using the notion of inclined restriction one arrives at τP;t = ι∗h t (T t ) , spat

or, pointwise, ∀x ∈ ∂𝜘 t (P) : τP;t (x) = ι∗h t,x (T t (x)) spat

spat

This is the generalization of the relation τ P;t = T t n in Euclidean mechanics.

7.5 Cauchy and Piola Stresses Let M = {𝜘 t }t∈𝕋 be a smooth motion and mat {(β mat t , τ P;t )}P∈Part(B),t∈𝕋 ,

spat

{(β t

spat

, τP;t )}P∈Part(B),t∈𝕋

(7.4)

7.5 Cauchy and Piola Stresses

|

187

be force systems. The Piola stress form (7.3) acts on the body B and the Cauchy stress form (7.4) acts on the shapes S𝜘t , t ∈ 𝕋. If P ∈ Part(B), then the powers (7.1) and (7.2) take the form mat F mat + ∫ V M;t ⌟ ι∗h (P t ) , t (V) = ∫ V M;t ⌟ β t ∂P

P spat F t (v)

= ∫

spat vt ⌟ βt

+

𝜘t (P)

v t ⌟ ι∗h t (T t ) .

∫ ∂𝜘t (P)

In the following, we omit inclined restrictions and simply write mat F mat + ∫ V M;t ⌟ P t , t (V) = ∫ V M;t ⌟ β t ∂P

P spat F t (v)

= ∫

spat vt ⌟ βt

+

𝜘t (P)

(7.5) ∫

vt ⌟ Tt .

∂𝜘t (P)

Stress forms can be considered as images of Hodge stars applied to fields of linear mappings. Choose a particular element 𝜘 ∈ M. Sections ∗ ∗ t ∈ Sec (ιS T P ⊗ T ∗ S𝜘 ) 𝜘

and p ∈ Sec(𝜘∗ T ∗ P ⊗ T ∗ B) .

correspond to the Cauchy and first Piola–Kirchhoff stress forms as follows [126]: ∗ ∗ T P ⊗ Λ n−1 (T ∗ S𝜘 )) , T ∈ Sec (ιS 𝜘 ∗ ∗

P ∈ Sec (𝜘 T P ⊗ Λ

n−1

∗

(T B)) ,

T = ∗2 t ,

(7.6)

P = ∗2 p ,

(7.7)

where ∗2 is the Hodge star operation applied to the “second leg”. In (7.6), the Hodge star is associated with the induced metric on the shape S𝜘 , and in (7.7), the Hodge star is associated with the material metric on B. Let t = t aβ e a ⊗ ̃e β , p = p aβ e a ⊗ E β , where (E α )nα=1 ⊂ Vec(B) and (̃e α )nα=1 ⊂ Vec(S𝜘 ) are frames on the body and the shape, respectively, and (e i )m i=1 ⊂ Vec(P) is a frame on P. Then in coordinates one has T = t aβ e a ⊗ (∗̃e β ) ,

P = p aβ e a ⊗ (∗E β ) ,

Since [123] n

∗E β = √G ∑ (−1)γ−1 G βγ E1 ∧ ⋅ ⋅ ⋅ ∧ ̂ Eγ ∧ ⋅ ⋅ ⋅ ∧ En , γ=1 n

∗̃e β = √g ∑ (−1)α−1 g βα ̃e1 ∧ ⋅ ⋅ ⋅ ∧ ̃êα ∧ ⋅ ⋅ ⋅ ∧ ̃e n , α=1

188 | 7 Stress Measures

we arrive at the following representations: n

T = ∑ (−1)α−1 g βα t aβ √g e a ⊗ (̃e1 ∧ ⋅ ⋅ ⋅ ∧ ̃êα ∧ ⋅ ⋅ ⋅ ∧ ̃e n ) ,

(7.8)

α=1 n

P = ∑ (−1)γ−1 G βγ p aβ √G e a ⊗ (E1 ∧ ⋅ ⋅ ⋅ ∧ ̂ Eγ ∧ ⋅ ⋅ ⋅ ∧ En ) .

(7.9)

γ=1

Note that the “exterior” part of the Cauchy stress form, the element of Ω n−1 (T ∗ S𝜘 ), is a differential form on the shape S𝜘 . In other words, it is an element of internal geometry of the shape. If n = dim B < m = dim P, then one does not identify spaces T x S𝜘 and T x P. The additional data is required to transform Ω n−1 (T ∗ S𝜘 ) into ∗ ∗ ∗ ∗ Ω m−1 (ιS T P). If m = n, then the spaces Ω n−1 (T ∗ S𝜘 ) and Ω m−1 (ιS T P) can be iden𝜘 𝜘 n tified. In this situation, one is able to change the frame (̃e α )α=1 into (e i )m i=1 .

7.6 Transformation from Spatial Description to Material Piola transformation. The Piola and Cauchy stress forms were introduced individually spat and p t . These powers are realizations of and appear in expressions for powers pmat t the total force in material and spatial descriptions. In this regard, they represent one and the same quantity and can differ by change of variables only. Let M = {𝜘t }t∈𝕋 be spat a smooth motion and the collections {τmat P;t }P∈Part(B),t∈𝕋 , {τ P;t }P∈Part(B),t∈𝕋 , be force systems. The body forces are absent. Fix t ∈ 𝕋. One can require that ∫ V M;t ⌟ P t = ∂P

∫

vt ⌟ Tt .

∂𝜘t (P)

̂ −1 ̂ t is an orientation-preserving diffeomorRecall that v t = V M;t ∘ 𝜘 t . Assume that 𝜘 phism⁴ for each instant t ∈ 𝕋. The change of variables theorem gives ̂ ∗t (v t ⌟ T t ) . ∫ V M;t ⌟ P t = ∫ 𝜘 ∂P

∂P

Let (e i )m i=1 ⊂ Vec(P) be a frame on P. Then, one has decompositions: v t = v it e i ,

T t = T iα e i ⊗ ω αt ,

where ω αt ∈ Ω n−1 (T ∗ S𝜘t ). One has v t ⌟ T t = v it T iα ω αt . Thus, ̂ ∗t ω αt . ̂ t ) (T iα ∘ 𝜘 ̂ t )ω αt = (v it ∘ 𝜘 ̂ t ) (e i ∘ 𝜘 ̂ t ) ⌟(T jα ∘ 𝜘 ̂ t ) (e j ∘ 𝜘 ̂t) ⊗ 𝜘 ̂ ∗t (v t ⌟ T t ) = (v it ∘ 𝜘 𝜘

4 A diffeomorphism is called orientation-preserving if its tangent map takes oriented bases of the domain to oriented bases of the codomain [3].

7.6 Transformation from Spatial Description to Material | 189

Introduce the operation that performs a pullback of the “geometrical part” of the covector-valued form: ∗ ̂ ∗t 2 : Sec (ιS T ∗ P ⊗ Λ n−1 (T ∗ S𝜘t )) → Sec (𝜘∗t T ∗ P ⊗ Λ n−1 (T ∗ B)) , 𝜘 𝜘 t

̂ t) ⊗ 𝜘 ̂ ∗t ω . ̂ ∗t 2 (ϑ ⊗ ω) := (ϑ ∘ 𝜘 𝜘

(7.10)

∗

̂ t is the composition of the mapping ̂ t 2 as the partial pullback. Here, ϑ ∘ 𝜘 We refer to 𝜘 ∗ ̂ t and the section ϑ of the cotangent bundle ιS T ∗ P. Using the operation (7.10) and 𝜘 𝜘 t

̂ t )(e i ∘ 𝜘 ̂ t ) gives the relation V M;t = (v it ∘ 𝜘

̂ ∗t 2 (T t ) . ̂ ∗t (v t ⌟ T t ) = V M;t ⌟ 𝜘 𝜘 Finally, ̂ ∗t 2 (T t ) . ∫ V M;t ⌟ P t = ∫ V M;t ⌟ 𝜘 ∂P

∂P

Since P and V M;t can be chosen arbitrarily, one obtains ∗

̂ t 2 Tt . Pt = 𝜘

(7.11)

This relation corresponds to the Piola transformation. The relations between stresses for the case when the body and physical space have Euclidean structure and dim B = dim P = 3 are illustrated in the diagram T∗P ∮∂P P

T ∗ P ⊗ Λ2 (T ∗ B) ∗2 T∗P ⊗ T∗B

∮𝜘(∂P)

p

B 𝜘 ∗2 Piola

T

T ∗ P ⊗ Λ2 (T ∗ P) t

∗2 T∗P ⊗ T∗P

The diagram schematically illustrates the relations P = ∗2 p and T = ∗2 t. Note that all mappings act from B. The geometrical sense for ∗2 is shown. Such an operator transforms T ∗ X ⊗ T ∗Y into T ∗ X ⊗ Λ2 (T ∗ Y). For economy, one symbol ∗2 is used here; but actually such mappings for B and P are different. The mapping 𝜘∗2 , as it is shown in the diagram, performs a pullback of the such part of the object that corresponds to the exterior 2-form in the physical space. This corresponds to the classical Piola transformation. If both the body and the physical space have Euclidean structure, then the integral of covector-valued form from T ∗ X ⊗ Λ2 (T ∗Y) is no more than the surface integral of covector density. The result of integration is a resulting contact force covector from T ∗ X, which is shown in the diagram.

190 | 7 Stress Measures Body force transformation. Let M = {𝜘t }t∈𝕋 be a smooth motion and the collections spat }t∈𝕋 , be force systems. The surface forces are absent. Fix t ∈ 𝕋. One {β mat t } t∈𝕋 , {β t can require that spat

∫ V M;t ⌟ β mat = ∫ vt ⌟ βt t

.

𝜘t (P)

P

̂ t is an orientation-preserving diffeomorphism for each instant Assuming again that 𝜘 t ∈ 𝕋 and using the change of variables theorem, one arrives at ̂ ∗t (v t ⌟ β t ∫ V M;t ⌟ β mat = ∫𝜘 t

spat

P

) .

P

We will use reasonings similar to Piola transformation. Let (e i )m i=1 ⊂ Vec(P) be a frame on P. Then one has decompositions: v t = v it e i ,

spat

βt

= e i ⊗ η t;i ,

where η t;i ∈ Ω n (T ∗ S𝜘t ). Thus, ̂ ∗t (v t ⌟ β t 𝜘

spat

̂ t ) (e i ∘ 𝜘 ̂ t ) ⌟ (e j ∘ 𝜘 ̂ t) ⊗ 𝜘 ̂ ∗t η t;j . ) = (v it ∘ 𝜘

Introduce the operation that performs a pullback of the “geometrical part”: ∗ ̂ ∗t 2 : Sec (ιS T ∗ P ⊗ Λ n (T ∗ S𝜘t )) → Sec (𝜘∗t T ∗ P ⊗ Λ n (T ∗ B)) , 𝜘 𝜘 t

̂ t) ⊗ 𝜘 ̂ ∗t ω . ̂ ∗t 2 (ϑ ⊗ ω) := (ϑ ∘ 𝜘 𝜘

(7.12)

Using the operation (7.12) results in ̂ ∗t (v t ⌟ β t 𝜘

spat

∗

spat

̂ t 2 (β t ) = V M;t ⌟ 𝜘

) .

Finally, we obtain ̂ ∗t 2 (β t = ∫ V M;t ⌟ 𝜘 ∫ V M;t ⌟ β mat t

spat

P

).

P

Since P and V M;t can be chosen arbitrarily, one obtains ∗

spat

̂ t 2 βt =𝜘 β mat t

.

(7.13)

The relation (7.13) is an equality between covector-valued n-forms. In this regard, one can obtain a coordinate form using the following proposition. Suppose that M and N are two smooth n-dimensional manifolds, and f : M → N is a smooth map. If (x i ) and (y i ) are local coordinates on M and N, respectively, then, [3] f ∗ (h dy1 ∧ ⋅ ⋅ ⋅ ∧ dy n ) = (h ∘ f) det (

∂y i ) dx1 ∧ ⋅ ⋅ ⋅ ∧ dx n . ∂x j

7.7 Example: Neo-Hookean Solids

|

191

Choose the local coordinates (Xα )nα=1 on B and (X β )nβ=1 on S𝜘t . Then η t;j = h t;j dX 1 ∧ ⋅ ⋅ ⋅ ∧ dX n . One has ̂ t ) det ( ̂ ∗t (η t;j ) = (h t;j ∘ 𝜘 𝜘

∂X α ) dX1 ∧ ⋅ ⋅ ⋅ ∧ dXn . ∂Xβ

Finally, one obtains the following representation for β mat t : ̂ t ) det ( = (h t;j ∘ 𝜘 β mat t

∂X α ̂ t ) ⊗ (dX1 ∧ ⋅ ⋅ ⋅ ∧ dXn ) . ) (e j ∘ 𝜘 ∂Xβ

7.7 Example: Neo-Hookean Solids Let t be defined by the expression t = −p0 I ♭p + μ B♭p , where μ is a constant, I is the identity tensor in the physical space, B is the left Cauchy–Green tensor, and p0 is a scalar-valued mapping (hydrostatic loading). This relation is analogous to the classical constitutive equation for an incompressible neoHookean material. Since I = ∂ X j ⊗ dX j ,

B = G im g lj F l m F k i ∂ X k ⊗ dX j ,

where (∂ X k ) and (dX k ) are coordinate frame and coframe in the physical space P, then I ♭p = g sj dX s ⊗ dX j ,

B♭p = G im g ks g lj F l m F k i dX s ⊗ dX j .

Thus, the coordinate representation for t has the form t = t sj dX s ⊗ dX j = {−p0 g sj + μG im g ks g lj F l m F k i } dX s ⊗ dX j . The corresponding covector-valued form is defined by the relation T = ∗2 t = t sj dX s ⊗ (∗dX j ) . One has that m

̂a ∧ ⋅ ⋅ ⋅ ∧ dX m , ∗dX j = √g ∑ (−1)a−1 g ja dX 1 ∧ ⋅ ⋅ ⋅ ∧ dX a=1

and thus, the following equality holds: m

T = √g ∑ (−1)a−1 g ja {−p0 g sj + μG im g ks g lj F l m F k i } dX s a=1

̂a ∧ ⋅ ⋅ ⋅ ∧ dX m ) . ⊗ (dX 1 ∧ ⋅ ⋅ ⋅ ∧ dX This relation shows that T depends on g and G.

192 | 7 Stress Measures Since P = 𝜘∗2 T, one has P = 𝜘∗2 T = 𝜘∗2 {t sj dX s ⊗ (∗dX j )} = t sj dX s ⊗ 𝜘∗ (∗dX j ) . Finally, one arrives at the formula for P: m

P = √g ∑ (−1)a−1 g ja {−p0 g sj + μG im g ks g lj F l m F ki } dX s a=1

̂a ∧ ⋅ ⋅ ⋅ ∧ dX m ) , ⊗ 𝜘∗ (dX 1 ∧ ⋅ ⋅ ⋅ ∧ dX ̂a ∧ ⋅ ⋅ ⋅∧ dX m ) is defined on the body B. where the exterior (n − 1)-form 𝜘∗ (dX 1 ∧ ⋅ ⋅ ⋅∧ dX

7.8 The Case dim B = dim P = 3 Obtain the explicit formula that allows one to calculate the Piola–Kirchhoff exterior stress form components by using the known Cauchy stress form, material and spatial metrics, and the configuration gradient. The particular case m = n = 3 is considered. As we will show, this formula looks similar to the classical one. We begin with the relation (7.8), which in our considerations has the form: T = t ab √g dX a ⊗ (g b1 dX 2 ∧ dX 3 − g b2 dX 1 ∧ dX 3 + g b3 dX 1 ∧ dX 2 ) .

(7.14)

To use (7.11) we need the formula for 𝜘∗ (dX i ∧ dX j ). From the definition of pullback, it follows that ∂𝜘 i ∂𝜘 j 𝜘∗ (dX i ∧ dX j ) = dXα ∧ dXβ . ∂Xα ∂Xβ Substituting formula (7.14) into (7.11), using the linearity of 𝜘∗ and the above formula for 𝜘∗ (dX i ∧ dX j ), we arrive at the relation P = J t ab √g g bk dX a ⊗ ((F −1 )1 k dX2 ∧ dX3 − (F −1 )2k dX1 ∧ dX3 + (F −1 )3 k dX1 ∧ dX2 ) , (7.15) ∂𝜘i α −1 −1 k where J = det( ∂Xα ), and F = (F ) k ∂Xα ⊗ dX is the inverse to F = T𝜘. Since comγ ponents for F −1 and its transpose, F −T , are related by the expression g ki (F −1 ) i = k k γβ −T −T G (F ) β , the relation (7.15) may be rewritten in terms of (F ) β . Now let us note that (7.9) in our particular case has the form: P = p aβ √G e a ⊗ (G β1 dX2 ∧ X3 − G β2 X1 ∧ X3 + G β3 X1 ∧ X2 ) . Then we obtain p aβ = √ where g = det[g ij ], and G = det[G ij ].

g b J t ab (F −T ) β , G

(7.16)

7.9 The Eshelby Energy-Momentum Tensor on Manifolds |

193

7.9 The Eshelby Energy-Momentum Tensor on Manifolds The Eshelby energy-momentum tensor can be generalized for the body B as follows. The balance equation on the body has the form: ♯

Div p♯ + ρ 0 b 0 = 0 .

(7.17)

Here, p is the first Piola–Kirchhoff stress tensor, p♯ = p aκ ∂ X a ⊗ ∂Xκ and⁵ Div p♯ = [∂ κ p aκ + p aκ Γ q

β κβ

+ p bκ F c κ γ a bc ]∂ X a ,

q

where Γ kl and γ kl are Levi–Civita connection coefficients on B and P, respectively. Applying contraction with F T = (F T )α j ∂Xα ⊗ dX j to (7.17) gives: ♯

F T ⌞ Div p + ρ 0 F T ⌞ b 0 = 0 .

(7.18)

For future needs, we define the bilinear operation ⌞⌞ , which satisfies the following property: ∂Xα ⊗ dX a ⊗ dXβ ⌞⌞∂ X b ⊗ ∂Xκ = ⟨dX a , ∂ X b ⟩ ⟨dXβ , ∂Xκ ⟩ ∂Xα . The following formula may be verified directly: Div(F T ⌞ p♯ ) = F T ⌞ Div p♯ + (∇F T )⌞⌞p♯ .

(7.19)

By (7.19) for the equation (7.18), we have ♯ Div(F T ⌞ p♯ ) − (∇F T )⌞⌞p♯ + ρ 0 F T ⌞ b 0 = 0 .

Subtracting the field⁶ (∇B W)♯ from the left and right-hand sides of the equation obtained, where W : B → ℝ is the elastic energy density defined at B, we obtain ♯ Div(F T ⌟ p♯ ) − (∇B W)♯ − (∇F T )⌞⌞p♯ + ρ 0 F T ⌞ b 0 = −(∇B W)♯ .

(7.20)

Let I denote the identity tensor field on B, I = ∂Xκ ⊗ dXκ . Since Div(WI ♯ ) = {G αβ ∂Xβ W} ∂Xα = (∇B W)♯ ,

5 According to Chapter 9, Section 9.2.5, the covariant derivative of a two-point tensor field Q = Q aκ ∂ Xa ⊗ ∂Xκ has the representation ∇Q = ∇β Q aκ ∂ Xa ⊗ ∂Xκ ⊗ dXβ , in which ∇β Q aκ = ∂ β Q aκ + Q aλ Γ κλβ + Q bκ F cβ γ abc . It follows that Div Q = ∇κ Q aκ . 6 Note that ∇B W is a covector field, and (∇B W)♯ is a vector field.

194 | 7 Stress Measures

the expression (7.20) takes the final form: ♯

Div e + (∇F T )⌞⌞p♯ − ρ 0 F T ⌞ b 0 = (∇B W)♯ ,

(7.21)

where e = WI ♯ − F T ⌞ p♯ is Eshelby energy-momentum tensor. ̃ be the elastic energy density represented as the function of the points and Let W ̃ fields defined on B: W(X) = W(X; F(X), . . . ). Following [8], define ∇B W|impl as⁷: ∇B W|impl = ∇B W − ∇B W|expl , where

󵄨 ̃ F(X0 ), . . . )󵄨󵄨󵄨󵄨X ∇B W|expl = ∇B W(X;

0 =X

.

In this case, the relation (7.21) may be rewritten in the form [8]: ♯

♯

♯

− Div e = fint + fext + finh , ♯

♯

♯

where fint is the internal force, fext is the body force, and finh is the Eshelby force. Such fields are represented by expressions ♯ ♯ fint = (∇F T )⌞⌞p♯ − ∇B W|impl , ♯

♯

fext = −ρ 0 F T ⌞ b 0 , ♯

♯

finh = −∇B W|expl .

󵄨 W(X; F(X0 ), . . . )󵄨󵄨󵄨󵄨X =X has the following sense: the covariant derivative is 7 The expression ∇B ̃ 0 ̃ taken from the field X 󳨃→ W(X; F(X0 ), . . . ), where F(X0 ), . . . are field values at a fixed point X0 . After calculation of the covariant derivative, all entries of X0 are replaced by X.

8 Material Uniformity and Inhomogeneity 8.1 Equivalence Relation Between Smooth Embeddings Suppose that M and N are smooth manifolds, s = dim M ≤ k = dim N. Let p ∈ M be a point. Establish an equivalence relation ∼p between smooth embeddings from C∞ (M; N) in such a way: for all smooth embeddings 𝜘1 , 𝜘2 ∈ C∞ (M; N), (𝜘1 ∼p 𝜘2 ) ⇔ ((𝜘1 (p) = 𝜘2 (p)) ∧ (T p 𝜘1 = T p 𝜘2 )) .

(8.1)

Denote an equivalence class by clp or by [𝜘]p . In coordinates, the relation (8.1) can be represented as follows: 𝜘1 ∼p 𝜘2 iff 𝜘1i (p1 , . . . , p s ) = 𝜘2i (p1 , . . . , p s ) ,

and

∂ j 𝜘1i |(p1 ,...,p s ) = ∂ j 𝜘2i |(p1 ,...,p s ) ,

for all i = 1, . . . , k, j = 1, . . . , s. Here, the mappings 𝜘1i , 𝜘2i are coordinate representations of 𝜘1 and 𝜘2 ; (p i ) are coordinates of p. The equivalence relation between diffeomorphisms of M to itself is defined similarly. Suppose that Φ : M → M is a diffeomorphism. Then, we apply¹ clΦ(p) ∘ [Φ]p := [𝜘 ∘ Φ]p , for some 𝜘 ∈ clΦ(p). Such a definition is well defined and is aligned with the composition of linear mappings. Moreover, the following properties hold: i) For any p ∈ M, one has clp ∘ [IdM ]p = clp ; ii) Suppose that p ∈ M is any point and Φ, Ψ are diffeomorphisms of M on itself. Then, clΦ∘Ψ(p) ∘ [Φ ∘ Ψ]p = (clΦ∘Ψ(p) ∘ [Φ]Ψ(p) ) ∘ [Ψ]p . This equality is true, since one has [𝜘 ∘ Φ ∘ Ψ]p = [𝜘 ∘ Φ]Ψ(p) ∘ [Ψ]p for 𝜘 ∈ clΦ∘Ψ(p).

8.2 Local Configurations and Simple Bodies Suppose that X ∈ B is a material point. Local configuration at X formalizes the idea of infinitesimal localization If the “observer” is placed into an infinitesimal neighborhood of the point X, then it can distinguish configurations only within this neighborhood. Assume that this localization has the first order: the “observer” cannot distinguish configurations that have the same values at X and the same linear approaches. The

1 The mapping 𝜘 ∘ Φ is a smooth embedding and [𝜘 ∘ Φ]p is understood in the sense of (8.1). https://doi.org/10.1515/9783110563214-008

196 | 8 Material Uniformity and Inhomogeneity

mathematical formalization of this idea can be provided as follows. The infinitesimal neighborhood of X is formalized as the tangent space TX B. The first-order localization is formalized by means of the equivalence relation ∼X (8.1) on the set C(B; P). Following W. Noll [104], we refer to each equivalence class [𝜘]X as a local configuration at X and denote it by KX . Let CX (B; P) be the quotient set (the set of all local configurations at X). The material at any point X ∈ B is characterized by its response. To obtain the response one uses measurers that give numerical data. Measured quantities at point X are represented mathematically by tensors. Among them may be scalar mappings (elastic energy density), second-rank tensors (Cauchy stress tensor). The ordered set of all such objects is denoted by R; its elements are called response descriptors. Assume that the response of the material at X depends on the first-order approximation of the configuration. That is, there is a mapping RX : CX (B; P) → R ,

CX (B; P) ∋ KX 󳨃→ RX (KX ) = r ∈ R .

If this mapping is specified for all X ∈ B, then the body B, endowed with the collection {RX }X of mappings, is called a simple body [104].

8.3 Material Uniformity Recall that in this book, we consider only simple bodies. Let X, Y ∈ B be material points. The notion of material isomorphism gives the precise meaning to the statement “material at X is the same as the material at Y.” More precisely, the points X and Y are materially isomorphic [6, 104], if there exists a diffeomorphism ΦXY : B → B, such that X = ΦXY (Y), and for all² KX ∈ CX (B; P), RX (KX ) = RY (KX ∘ [ΦXY ]Y ) . The equivalence class [ΦXY ]Y is called a material isomorphism from TY B to TX B. We call the mapping ΦXY a material automorphism from Y to X. One has the following properties: a) ΦXX = IdB is a material automorphism from X to X. b) If ΦXY is a material automorphism from Y to X, ΦYZ is a material automorphism from Z to Y, and then ΦXY ∘ ΦYZ is a material automorphism from Z to X. c) If ΦXY is a material automorphism from Y to X, then Φ−1 XY is a material automorphism from X to Y.

2 Note that the composition KX ∘ [Φ XY ]Y is a local configuration at Y.

8.3 Material Uniformity

|

197

Remark 8.1. Here, we will give the sketch of the proof of properties a)–c). a) For any KX ∈ CX (B; P) according to property i), Section 8.1, one has KX ∘ [IdB ]X = KX . Here, ΦXX = IdB . b) For any KX ∈ CX (B; P), using property ii) from Section 8.1, one obtains KX ∘ [ΦXY ∘ ΦYZ ]Z = (KX ∘ [ΦXY ]Y ) ∘ [ΦYZ ]Z ∈ CZ (B; P) , and RX (KX ) = RZ ((KX ∘ [ΦXY ]Y ) ∘ [ΦYZ ]Z ). c) For any KY ∈ CY (B; P), applying properties a) and b), one obtains −1 RY (KY ) = RY ((KY ∘ [Φ−1 XY ] ) ∘ [ΦXY ]Y ) = RX (KY ∘ [ΦXY ] ) , X

X

and this implies that Φ−1 XY is a material automorphism from X to Y. Remark 8.2. Suppose that the physical space P is a Euclidean affine space E. A local configuration KX can be identified with a linear mapping K X : TX B → V as follows. Choose some configuration 𝜘 ∈ KX and define K X by K X := TX 𝜘. According to the definition of the relation ∼X , the mapping K X does not depend on the choice of a representative 𝜘 and is fully determined by the class KX . In the case of the physical spaceE, we identify a local configuration KX with the corresponding linear mapping K X . A material isomorphism can be defined as a vector space isomorphism ΦXY : TY B → TX B. Such a definition corresponds to the conventional approach [104]. Suppose that all points of the body B are materially isomorphic to each other. That is, the material (response) at any two points X, Y ∈ B is the same. Such a body is called materially uniform. From now, we consider materially uniform simple bodies. Physically, a materially uniform simple body consists of infinitesimal representative volumes with the following property. Each of them, if separated from others and placed in a “mechanical test set up”, gives the same response on the same deformation. Now assume that one fixed a point X0 ∈ B and chosen some local configuration KXR0 . Let {ΦX0 X }X∈B be a family of material automorphisms. For points X ∈ B, X ≠ X0 , denote KXR := KXR0 ∘ [ΦX0 X ]X . Thus, one has the family Ref B = {KXR }X∈B of local configurations. Such local configurations give the same response: ∀X ∈ B :

RX (KXR ) = RX0 (KXR0 ) .

The family Ref B = {KXR }X∈B depends on a family of material automorphisms and a point X0 . We refer to the family Ref B as the uniform reference³. Let X00 ∈ B be a point, such that X00 ≠ X0 . The family Ref B = {KXR }X∈B can be represented in terms of X00 : if X ≠ X0 , then KXR = KXR00 ∘ [ΨX00 X ]X , where ΨX00 X = Φ−1 X0 X00 ∘ ΦX0 X . The above properties a)–c) were used for this relation.

3 This corresponds to the classical Nolls definition [104].

198 | 8 Material Uniformity and Inhomogeneity

Fig. 8.1: A materially uniform body.

R Suppose that Ref B = {KXR }X∈B . Choosing for any X ∈ B a representative 𝜘X ∈ R one arrives at the family Ref B = {𝜘X }X∈B . One observes the family {S𝜘R }X∈B of X shapes and splits them into elementary volumes. Volumes that correspond to every R point 𝜘X (X), give the same response. However, if one fixes a point X and considers R (Y), then their responses do not coincide with the elementary volumes at points 𝜘X response at X, in general. These reasonings are illustrated in Figure 8.1. A uniform reference Ref B = {KXR }X∈B generates a field of m × n matrices of rank n. Let X ∈ B. Choosing 𝜘 ∈ KXR , apply

KXR ,

K iβ (X) :=

∂𝜘i 󵄨󵄨󵄨 󵄨󵄨 , ∂Xβ 󵄨󵄨φ(X)

where 𝜘 i is a coordinate representation of 𝜘, (Xβ ) are local coordinates on B, and φ is a coordinate mapping on B. By virtue of the definition, the matrix [K i β (X)] does not depend on a representative 𝜘. Thus, the family Ref B gives the family Ref B = {[K i β (X)]}X∈B of matrices. If there exists a configuration 𝜘 R ∈ C(B; P) such that 𝜘 R ∈ KXR for every X ∈ B, then each elementary volume of the shape S𝜘R gives the same response. Such a body

Fig. 8.2: A homogeneous body.

8.4 A Material Metric

| 199

Fig. 8.3: An inhomogeneous body.

is called homogeneous. If such a configuration does not exist, then the body is called inhomogeneous. Figures 8.2 and 8.3 illustrate the difference between homogeneous and inhomogeneous bodies. Cubes represent uniform shapes.

8.4 A Material Metric In reasonings provided in the previous chapters, a body as a manifold is endowed by some abstract metric G. Despite of the spatial metric g, which is given a priori, the material metric is not presupposed to be defined independently of the physical process. For a materially uniform simple body, the material metric can be constructed in terms of a uniform reference. Choose a uniform reference Ref B = {KXR }X∈B and construct the family Ref B = R {𝜘X }X∈B . Using elements of the family Ref B one can obtain the collection {(B, R ∗ G(X) )}X∈B of Riemannian spaces with metrics G(X) := (𝜘X ) g that are pullbacks R of the physical metric g, relative to the embeddings 𝜘X . Thus, ∀Y ∈ B ∀u, v ∈ TY B :

R R G(X) |Y (u, v) = g (TY 𝜘X (u), TY 𝜘X (v)) .

The corresponding Levi–Civita connection coincides with the Levi–Civita connection on⁴ P. Now one can synthesize the Riemannian metric, which is non-Euclidean in general. Assume that the section G : B → T ∗ B ⊗ T ∗ B, defined by the formula GX := G(X) |X , is smooth. Each of the values G(X) |X represents a symmetric positive definite bilinear form. Hence, G is a Riemannian metric. Since it is defined on the body B, we refer to G

4 In the particular case of P = E, we have that the considering connection is Euclidean. In this case, the pullback transformation introduces some specific curvilinear coordinates on B.

200 | 8 Material Uniformity and Inhomogeneity

as a material metric. By the definition, the material metric is combined from the family of Riemannian metrics {G(X) }X∈B , i.e., ∀X ∈ B

R R GX (u, v) = g (TX 𝜘X (u), TX 𝜘X (v)) .

∀u, v ∈ TX B :

(8.2)

The material metric (8.2) depends on the uniform reference Ref B , but does not depend on certain representatives from each equivalence class. This motivates one to rewrite (8.2) in terms of Ref B as follows: ∀X ∈ B ∀u, v ∈ TX B :

GX (u, v) = g (KXR (u), KXR (v)) .

(8.3)

Here, KXR (u) := TX 𝜘(u), for some 𝜘 ∈ KXR . Remark 8.3. If the physical space is affine Euclidean, E, then (8.3) is represented in R terms of K R : X 󳨃→ K X in the following way: ∀u, v ∈ TB :

G(u, v) = [K R (u)] ⋅ [K R (v)] .

This is the classical formula that was introduced by Noll [104]. Since the body B is endowed with the material metric G, one can introduce the Levi– Civita connection on it. The connection coefficients are defined via the formula Γ αβγ = where G αβ = g sp K sα K

p β

G ατ (∂ β G γτ + ∂ γ G βτ − ∂ τ G βγ ) , 2

and K iβ =

∂𝜘i . ∂Xβ

8.5 Bodies with Variable Material Composition 8.5.1 Formalization of Bodies with Variable Material Composition Suppose that material manifold B can vary in time. This means that material points can be added to or removed from B. This situation arises, for example, in the modeling of growing solids [25, 127–129], biological tissues [130] the processes of freezing, melting, and ablation, etc. Such varying in time bodies must be characterized not only by their geometrical structure, but also by the process of their variation in time. In the following, we will refer to this process as the evolutionary process. Bodies with variable material composition can be represented by a family G = {Bα }α∈𝕀 of n-dimensional bodies. Here, 𝕀 is a linearly ordered set of indices. For each index α ∈ 𝕀, the body Bα characterizes an instantaneous material composition of the solid. Assume that 𝕀 is a subset of ℝ. The number Card(𝕀) characterizes the type of evolutionary process. When Card(𝕀) = N < ∞, the process may be qualified as discrete, while Card(𝕀) = ℵ1 corresponds to a continuous process.

8.5 Bodies with Variable Material Composition

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Assume that during the evolutionary process no material can be removed. That is, family G is ordered by the subset relation, i.e., ∀α, β ∈ 𝕀 :

(α < β) ⇒ (Bα ⫋ Bβ ) .

Without loss of generality one may assume that 𝕀 = {1, . . . , N} in the case of the discrete process and 𝕀 = [a0 , a1 ] in the case of the continuous one. Thus, the sets B∗ = ⋂ B α , α∈𝕀

B∗ = ⋃ B α α∈𝕀

are non-empty and represent the initial and final bodies, respectively. They coincide with some uniquely determined bodies from G. The body B∗ represents a substrate, while the body B∗ represents the total body composition. Following technological interpretation we refer to each element of G as an assembly. For each pair of indices α, β ∈ 𝕀, the corresponding bodies Bα , Bβ have independent topological structures, in general. In particular, they have different topological invariants. Figure 8.4 shows the evolution of a body that is topologically equivalent to the n-solid torus, during which the Euler characteristic χ changes due to the attachment of extra material. In this book, we consider a particular case when Bα is a “part” of Bβ , if α < β. It can be formalized as follows: if α < β, then Bα is an embedded submanifold of Bβ . Thus, for each α ∈ 𝕀, the space Bα is an embedded submanifold of B∗ . Since dim Bα = dim B∗ , this implies that Bα is an open subset of B∗ . In the case of the continuous process, we wish to represent each element of the family G that corresponds to α ∈ 𝕀, as Bα = B∗ ∨ ̃ Bα . Here, the symbol ̃ Bα represents an extra body, formed by extra material that flowed in to the substrate B∗ = Ba0 . In

k = 1, χ(Ω1 ) = n

k = 2, χ(Ω2 ) = n + 2

k = 3, χ(Ω3 ) = n + 4

k = 4, χ(Ω4 ) = n + 6

Fig. 8.4: Change in the topological structure of the body in the process of its evolution.

202 | 8 Material Uniformity and Inhomogeneity the case of discrete process, we want the following expression: Bs+1 = Bs ∨ Ls . Here, the symbol Ls corresponds to the layer that is attached to the previous assembly, Bs . In both cases, the symbol ∨ denotes the operation of joining, which assigns to each pair of bodies a new body, which contains both of them. Although the partition of the body into layers and its assembling seems to be natural operations on the set of material points, constituting the body, their formal implementations have technical issues. In particular, the set difference of the body and its part results in a set, which is not open and, therefore, cannot be a body. Moreover, after the union operation applied to two bodies with a partially coinciding boundary, the intersection of boundaries would not be a part of the result (recall that we treat a body as an open set). Intuitive notions of an extra body (or layer, in the discrete case) and the joining operation may be formalized as follows. Denote the topology of B∗ by S. Since each n-dimensional body contained in B∗ is an open set, and vice versa, it is sufficient to define the operation of the detachment of an extra body (or layer) that acts on open set, represented a body, and results in open set, represented a part of this body. Let us consider the case when one body (or “substrate”) is contained in another. Let B1 and B2 be n-dimensional smooth submanifolds of B∗ (that is, bodies), and B1 ⊂ B2 . Then, the set B1 is open in the topological space B2 . Therefore, the difference B2 \B1 represents a closed set relative to the induced topology SB2 of B2 . Hence, one has that B2 \ B1 = B2 \ B1 , and we arrive at the following representation: B2 \ B1 = IntB2 (B2 \ B1 ) ∪ ∂B2 (B2 \ B1 ) , where the topological operations of closure, interior (IntB2 ), and boundary (∂B2 ) are considered in the topology SB2 . The set IntB2 (B2 \ B1 ) is open in B∗ . Since the interior operation is defined as the union of all open sets contained in the given set, we have that IntB2 (B2 \ B1 ) = Int(B2 \ B1 ) , where Int is the interior operation relative to the topology of B∗ . These reasonings allow one to introduce the operation J of layer selection, which for B1 ⊂ B2 returns the set J(B2 , B1 ) := Int(B2 \ B1 ) , to which we refer as the layer or extra body (in the continuous case). The operation J allows one to partition any n-dimensional body B2 into a body B1 ∈ SB2 and the body J(B2 , B1 ). The symbol ∨ indicates this partition and is referred to as the joining operation: B2 = B1 ∨ J(B2 , B1 ) := B1 ∪ Int(B2 \ B1 ) ∪ ∂B2 (B2 \ B1 ) .

8.5 Bodies with Variable Material Composition

∙ ∙

|

203

Finally, ̃ α , where B ̃ α := J(Bα , B∗ ). For a continuous process, one has Bα := B∗ ∨ B For a discrete process, Bs+1 := Bs ∨ Ls , in which Ls = J(Bs+1 , Bs ). In addition, since no material is removed, B∗ = B1 . Thus, in this case, B∗ = BN = (. . . (B∗ ∨ L1 ) ∨ ⋅ ⋅ ⋅ ∨ LN−2 ) ∨ LN−1 ,

L0 := B∗ .

Remark 8.4. The operation ∨ is not equal to the conventional set union and has the following physical sense, which we will explain in the case of the discrete process. When we obtain a sequentially new assembly Bs+1 from the previous assembly Bs and the layer Ls , we have to include all interior points of Bs and Ls and, in addition, such points of common boundary of Bs and Ls that represent a thin film between “gluing” solids. Remark 8.5. Consider an example. Suppose that one has a discrete process and ̃ Int [Bs+1 \ Bs ] := Int (Bs ∪ Int [Bs+1 \ Bs ]) , Bs ∨ where the closure and interior (Int) are taken in B∗ = BN . In the simple case one can use such a definition, but in general it may give an erroneous result. Let B∗ = ℝ2 , the set Bs+1 be an open disk with unit radius, and the center punctured at zero, and the set Bs be an open disk with radius 1/2 and the center punctured at zero. In this case, ̃ Int [Bs+1 \ Bs ], but at the same time, Bs+1 = Bs ∨ Int [Bs+1 \ Bs ]. Bs+1 ≠ Bs ∨ In the examples and the remainder of the chapter, we will consider the case when the physical space is Euclidean, E.

8.5.2 The Discrete Process Suppose that a discrete evolutionary structure is represented by a family G = {Bs }Ns=1 . Assume that each element of G is a materially uniform inhomogeneous simple threedimensional body. Then it can be represented by the following decomposition: Bs = (. . . (B∗ ∨ L1 ) ∨ ⋅ ⋅ ⋅ ∨ Ls−2 ) ∨ Ls−1 ,

s ∈ {1, . . . , N} .

(8.4)

Each layer Lk is assumed to be a materially uniform simple body that has a stress-free shape in E. Moreover, assume that for each s ∈ {1, . . . , N} and any k ∈ {0, . . . , s − 1}, there exists a configuration 𝜘Rs,k ∈ C(Bs ; E) of the body Bs such that 𝜘 Rs,k (Lk ) is a uniform shape of Lk . For the body B4 such an assumption is illustrated in the diagram:

204 | 8 Material Uniformity and Inhomogeneity

The areas in dark color correspond to a uniform shape.

8.5.3 The Continuous Process Evolving bodies. Consider the case when material flows continuously into a substrate. This case might match the models for a variety of technological processes, like LbL processing, stereolithography, chemical vapor deposition (CVD), physical vapor deposition (PVD) and others, when the number of successive depositions of thin layers is so big that the whole process, despite of its step-by-step character, is similar to a continuous one. Such an evolving body can be represented by a continuous family G = {Bα }α∈[a0 ,a1 ] of three-dimensional materially uniform simple bodies. Each element of the family contains the substrate B∗ = Ba0 as a subbody, which will be referred to as the initial body. With the operation ∨ one can decompose an element Bα ̃ α as Bα = B∗ ∨ ̃ of the family G into the initial body B∗ and an extra body B Bα . The ̃ = {B ̃ α }α∈]a ,a ] . collection of all extra bodies constitutes the family G 0 1 Since B∗ represents a substrate, its shape at the beginning of the deposition process may be regarded as uniform (free from residual stresses). However, it cannot be said about any shape of extra bodies, because various physical and chem-

8.5 Bodies with Variable Material Composition

| 205

ical processes that accompanied the deposition, cause local shrinkage. Denote⁵ ̃∗ = J(B∗ , B∗ ) = ⋃ ̃ B α∈]a 0 ,a 1 ] B α . Such a body contains all extra material that was continuously added to the substrate B∗ during the whole evolving process. With a view to the step-by-step microstructure of the technological process one can accept ̃∗ as a three-dimensional manifold has the structure of the additional assumption: B ̃∗ can be a foliated manifold with 2D foils, treated as material surfaces [110]. Thus, B endowed with the structure of the principal bundle over the one-dimensional base. ̃∗ If, beyond that, each foil is diffeomorphic to some smooth 2D manifold, then B can be endowed with the structure of a fiber bundle with a typical fiber F. These considerations can be stated more strictly in the following way: ̃∗ , 𝕁, π, F) with the total space ̃ (i) There exists a smooth fiber bundle (B B∗ , the onedimensional base 𝕁, and the projection π : ̃ B∗ → 𝕁, which is smooth and surjec−1 tive. Each fiber S i := π ({i}), i ∈ 𝕁, is diffeomorphic to the typical fiber F, which is a smooth two-dimensional manifold. ̃∗ , 𝕁, π, F) is related with the family G ̃ as follows. For each (ii) The fiber bundle (B α ∈]a0 , a1 ] there exists a submanifold 𝕁α ⊂ 𝕁, such that⁶ ̃α = ⋃ Si . B i∈𝕁α R R ̃∗ ; E) with (iii) There exists a family Ref B ̃ ∗ = {𝜘X }X∈B ̃ ∗ of configurations 𝜘X ∈ C(B ̃∗ and the following property. If X ∈ B ̃∗ , then each point from a the index set B R two-dimensional submanifold 𝜘X (S π(X) ) in E has some uniform spatial neighR R borhood. Moreover, if X, Y ∈ Si for some i ∈ 𝕁, then 𝜘X = 𝜘Y .

Assumption (i) asserts that ̃ B∗ is represented as a union of disjoint two-dimensional manifolds Si : ̃∗ = ⋃ S i . B i∈𝕁

̃∗ , 𝕁, π, F) in accordance with foliated structure Assumption (ii) set the fiber bundle (B ̃ of the elements of G. Indeed, for each α ∈]a0 , a1 ], there exists a smooth fiber bundle ̃ α , 𝕁α , π α , F) with the one-dimensional base 𝕁α and the projection π α : B ̃ α → 𝕁α . (B ∗ ̃ , 𝕁, π, F). Assumption (iii) formalizes the sugSuch a bundle is the restriction of (B gestion related with the technological process of deposition. This claims that for every point X ∈ ̃ B∗ , there exists a configuration that brings the neighborhood of the point X, together with neighborhoods of all points from the fiber to which X belongs, to the uniform state. ̃ α denote 𝜘 R := 𝜘 R | ̃ . Let α ∈]a0 , a1 ]. For each material point X ∈ B α;X X Bα

̃α ⊂ B ̃∗. 5 For each α ∈]a0 , a1 ] one has B 6 Here 𝕁a1 := 𝕁.

206 | 8 Material Uniformity and Inhomogeneity Intermediate configuration. Fix α ∈]a0 , a1 ] and let 𝜘∘α ∈ C(Bα ; E) be a configuration. The shape 𝜘∘α (Bα ) may serve as a “model” of the body Bα in the following sense. Any coordinate net of this shape can be transferred to the body Bα . Figuratively speaking, the body is “dressed up” with a coordinate net, taken from the shape 𝜘∘α (Bα ). Consider this in detail. Choose local coordinates (ξ α )3i=1 on the shape 𝜘∘α (Bα ). Let H ∘ : U → ℝ3 be a corresponding coordinate map on some open set U ⊂ 𝜘∘α (Bα ). Then, the composition ̂ ∘α |(̂𝜘∘α )−1 (U) is a coordinate map on the open set (̂ 𝜘∘α )−1 (U) of Bα . The correspondH∘ ∘ 𝜘 ing frame and coframe fields on the body are denoted by (∂ ξ αi )3i=1 , and, consequently, (dξ αi )3i=1 . Despite the fact that local coordinates on the body and its intermediate shape have the same notations, the corresponding frames and coframes are different, see Tab. 8.1. We will refer to the configuration 𝜘∘α as intermediate configuration and to the shape 𝜘∘α (Bα ) as intermediate shape. Tab. 8.1: Correspondence between frames and coframes on the body and the intermediate shape. Bα

𝜘∘α (Bα )

Coordinate map

̂ ∘α : X 󳨃→ (ξ α1 , ξ α2 , ξ α3 ) H∘ ∘ 𝜘

H ∘ : x 󳨃→ (ξ α1 , ξ α2 , ξ α3 )

Coordinates

(ξ α1 , ξ α2 , ξ α3 )

(ξ α1 , ξ α2 , ξ α3 )

Frame

(∂ ξ i )

Coframe

3

3

(e ξ i )

i=1 i 3 (dξ α )i=1 α

α

ξ αi

(e )

i=1 3 i=1

Parameterized intermediate configuration. Let an intermediate configuration be such that the corresponding deformations from the intermediate shape to locally stress-free reference shapes would depend on scalar parameters, which represent the unknown parameters of the model. Such parameters (more precisely, smooth functions), should be obtained from the evolutionary problem. Thus, we assume that for each α ∈]a0 , a1 ], there exists a configuration 𝜘∘α ∈ C(Bα ; E), such that 1) the shape 𝜘∘α (B∗ ) is uniform; ̃ α , the deformation 2) for each X ∈ B −1 ∘ ̃ R ̃ ̃γ α;X = 𝜘 ̂ Rα;X ∘ (̂ 𝜘∘α |B ̃ α ) : 𝜘 α (B α ) → 𝜘 α;X (B α ) ,

̃ α ) into the shape 𝜘 R (B ̃ α ) is prescribed and depends on k ≥ 1 from the shape 𝜘∘α (B α;X 1 k free scalar parameters ρ α;X , . . . , ρ α;X ∈ ℝ; 3) the mappings ̃ jα : B ̃α → ℝ , b

̃ jα : X 󳨃→ ρ j , b α;X

are smooth for all j = 1, . . . , k and α ∈]a0 , a1 ]. j Since the deformation ̃γ α;X is prescribed up to parameters ρ α;X , the family R ̃ }α∈]a ,a ] of uniform references can be obtained in a chart, induced by the interme{K α 0 1

8.5 Bodies with Variable Material Composition

| 207

̃ α ). diate configuration. Now, define local coordinates (Ξ iα;X )3i=1 on each shape 𝜘Rα;X (B R ̃ α ) is represented by Thus, the deformation gradient T y ̃γ α;X at point y ∈ 𝜘 α;X (B 󵄨 ∂ ̃γ iα;X (ξ α1 , ξ α2 , ξ α3 ; b 1α (X), . . . , b kα (X)) 󵄨󵄨󵄨 󵄨󵄨 e i | ⊗ e ξ αj | ̃ γ α;X (y) , 󵄨󵄨 Ξ α;X y j ∂ξ α 󵄨󵄨󵄨y

T y ̃γ α;X =

where ̃γ iα;X is the coordinate representation of ̃γ α;X . For any point X ∈ Bα , the configuration gradient TX 𝜘∘α : TX Bα → V has the dyadic decomposition TX 𝜘∘α = e ξ αi |x ⊗ dξ αi |X , ̃α where x = 𝜘∘α (X). Thus, the tangent map TY 𝜘 Rα;X : TY ̃ Bα → V to 𝜘 Rα;X at point Y ∈ B R ∘ ̂ α |B can be derived as a composition TY 𝜘 α;X = T y ̃γ α;X ∘ TX 𝜘 ̃α : TY 𝜘 Rα;X

̃ 1 (X), . . . , b ̃ k (X)) 󵄨󵄨󵄨 ∂ ̃γ iα;X (ξ α1 , ξ α2 , ξ α3 ; b 󵄨󵄨 α α 󵄨󵄨 e Ξ i |y ⊗ dξ αj |Y , = j α;X 󵄨󵄨󵄨 ∂ξ α 󵄨y

̃ : X 󳨃→ TX 𝜘 R is defined at point where y = 𝜘 Rα;X (Y). Hence, the uniform reference K α α;X ̃ α by the following expression: X∈B R

̃ R |X = K α

̃ 1 (X), . . . , b ̃ k (X)) 󵄨󵄨󵄨 ∂ ̃γ iα;X (ξ α1 , ξ α2 , ξ α3 ; b 󵄨󵄨 α α 󵄨󵄨 e Ξ i |x ⊗ dξ αj |X , j 󵄨󵄨 α;X ∂ξ α 󵄨󵄨x

(8.5)

where x = 𝜘Rα;X (X). ̃ jα , j = The material metric. So, uniform reference was determined up to mappings b 1, . . . , k. It is convenient to extend them over the whole Bα , i.e., to construct the mapj ping b α : Bα → ℝ defined by the relation {0, j b α (X) = { j ̃ (X), b { α

X ∈ B∗ ∪ ∂Bα (Bα \ B∗ ) , ̃α . X∈B

The set ∂Bα (Bα \ B) represents the surface of discontinuity, which appears between the substrate and the extra body. It follows from the homogeneity of the body B∗ that its material Riemannian metric is Euclidean. Thus, in the general case, the material metric in the whole body Bα would have points of discontinuity in ∂Bα (Bα \ B). Denote the metric on E by g, then {(𝜘∘ )∗ g(X), Gα (X) = { α ̃ (X), G { α

X ∈ B∗ ∪ ∂Bα (Bα \ B∗ ) , ̃α , X∈B

(8.6)

̃α is defined by (8.3). represents the total material metric Gα in Bα . Here, G j The functions b α , j = 1, . . . , k, on which (8.5) and (8.6) depend, define the distortion parameters. They can be defined by the modeling of physical and chemical

208 | 8 Material Uniformity and Inhomogeneity

processes during adhering of the layer. One of the most simple factors defining such processes is shrinkage, which occurs during a relatively short time interval (approximately instantly in the time scale of the whole adhering process). The idea of instantaneous shrinkage leads to the following formalization: j

j

∀α < β ∈]a0 , a1 ] : b β |B ̃α = bα ,

j = 1, . . . , k .

These equalities mean that parameters governing inhomogeneity in the solid cannot vary in internal parts of it.

9 Material Connections Introductory remarks. Let M be a smooth manifold. For a smooth vector field u ∈ Vec(M), the partial derivatives ∂ i u j do not form the components of a second-order tensor, in general. Similarly, for a 1-form ν ∈ CVec(M), the partial derivatives ∂ i ν j do not form any tensor, although the differences ∂ i ν j − ∂ j ν i form the exterior derivative dν. The notion of derivation needs to be generalized, since we are dealing with a body B and a physical space P, which are not Euclidean in general, and intend to formulate differential balance equations. This generalization is provided by covariant differentiation. Let us explain the difficulty of introducing the covariant derivative straightforwardly, as a limit of the relative increment. Let (E, M, π) be some smooth vector bundle over a smooth manifold M and s : M → E be its section. If p, q ∈ M are different material points, then the sum s p + s q has no meaning. Indeed, for p ≠ q, the values s p and s q belong to the non-intersecting vector spaces Ep and Eq . From this follows the difficulty of defining the derivative via a limit of the relative increment. One needs an extra intrinsic rule to connect Ep and Eq . Such a rule is provided with a connection on the vector bundle. In Chapter 14, we introduce the operation of covariant differentiation, drawing on a very general notion of the principal bundle. Since a vector bundle is a particular case of the principal bundle, the reasonings of aforementioned chapter can be directly applied to it. However, the operation of covariant differentiation on vector bundles can be constructed axiomatically. This is realized in this chapter. In Chapter 8, the material metric G was introduced. This metric, in turn, generates the Levi–Civita connection on the body. The Riemann curvature tensor of such a connection gives a measure of inhomogeneity, the scalar curvature. In this chapter, we provide a second way of constructing the connection on the tangent bundle of the body. This way is intimately related with Cartan’s idea of the moving frame method, and as a result, allows one to construct a uniform shape that would be a non-Euclidean space.

9.1 Connections on Vector Bundles Axiomatic definition. The connection ∇ on a smooth vector bundle (E, M, π) is a mapping ∇ : Vec(M) × Sec(E) → Sec(E) , (u, s) 󳨃→ ∇u s , which satisfies the following axioms [61, 131, 132]: (∇1 ) ∀u ∈ Vec(M) ∀s1 , s2 ∈ Sec(E) : ∇u (s1 + s2 ) = ∇u s1 + ∇u s2 ; (∇2 ) ∀u 1 , u 2 ∈ Vec(M) ∀s ∈ Sec(E) : ∇u1 +u2 s = ∇u1 s + ∇u2 s;

https://doi.org/10.1515/9783110563214-009

210 | 9 Material Connections (∇3 ) ∀u ∈ Vec(M) ∀f ∈ C ∞ (M) ∀s ∈ Sec(E) : ∇u (fs) = u(f)s + f∇u s; (∇4 ) ∀u ∈ Vec(M) ∀f ∈ C ∞ (M) ∀s ∈ Sec(E) : ∇fu s = f∇u s. We refer to ∇u as a covariant derivative along u. The Leibniz rule (∇3 ) and the property (∇4 ) imply that ∇u satisfies the locality property. That is, let U ⊂ M be any open set. Suppose that vector fields u 1 , u 2 , and sections s1 , s2 of E are equal in U: u 1 |U = u 2 |U and s1 |U = s2 |U . Then, ∇u s1 | U = ∇u s2 | U . This property allows one to define connection¹ [131] ∇|U : Vec(U) × Sec(E|U ) → Sec(E|U ) on an open set U ⊂ M by using the original connection ∇. We refer to ∇|U as a restriction of ∇ and in the following, we omit the character U after the vertical bar. Coordinate representation. Let s ∈ Sec(E) and u ∈ Vec(M), while U ⊂ M be a coordinate domain. Choose a smooth local frame (e a ) for E over U, and coordinate frames (∂ c ), (dx c ), respectively, for TM, T ∗ M over U. Now obtain coordinate representations for the restriction of ∇u s on U. Firstly, note that there exist smooth mappings ω bca ∈ C∞ (U), such that ∇∂ a e b = ω c ab e c . (9.1) Then, applying the Leibniz rule (∇3 ), we arrive at the expression ∇u s = u a {∂ a s c + s b ω c ab } e c , where u = u a ∂ a , s = s b e b . The covariant derivative of tensorial fields. Given is that connection ∇ on a smooth vector bundle can be extended to the other levels of the tensorial “tower” inductively. First, define the mapping ∇sc : Vec(M) × C∞ (M) → C∞ (M) ,

∇sc u f := u(f) .

Next, define the connection ∇∗ : Vec(M) × Sec(E∗ ) → Sec(E∗ ) on the dual vector bundle (E∗ , M, π) by the following characteristic property: ∀s ∈ Sec(E) ∀g ∈ Sec(E∗ ) ∀u ∈ Vec(M) :

∗ ∇sc u ⟨g, s⟩ = ⟨∇ u g, s⟩ + ⟨g, ∇ u s⟩ .

1 Here, E|U is the restriction of vector bundle E, i.e., the smooth vector bundle with base U and total space E|U = ⨆p∈U Ep .

9.1 Connections on Vector Bundles |

211

Let U ⊂ M be a coordinate domain, (e a ) be a smooth local frame for E over U, and (e a ) be its dual. Introduce the notations u = ∂ a , g = e b , s = e c . Then, in U one has ∇∗∂ a e b = ν b ad e d , where ν b ad ∈ C∞ (U) are unknown functions. Since ⟨e b , e c ⟩ = δ bc , with (9.1), one obtains ν b ac + ω b ac = 0 , which implies ∇∗∂ a e b = −ω b ad e d .

(9.2)

For a smooth section g ∈ Sec(E∗ ) and a vector field u ∈ Vec(M), the representation ∇∗u g = u a {∂ a g c − g b ω b ac } e c , is a direct consequence of (9.2) and the Leibniz rule (∇3 ). Here, g = g a e a , u = u b ∂ b . Suppose now that the connections ∇1 and ∇2 are defined on smooth vector bundles (E1 , M, π1 ) and (E2 , M, π2 ) over the same base M. These connections induce new ones, ∇E1 ⊕E2 : Vec(M) × Sec(E1 ⊕ E2 ) → Sec(E1 ⊕ E2 ) , ∇E1 ⊗E2 : Vec(M) × Sec(E1 ⊗ E2 ) → Sec(E1 ⊗ E2 ) , which are defined on the direct sum (E1 ⊕ E2 , M, πE1 ⊕E2 ) and the tensor product (E1 ⊗ E2 , M, πE1 ⊗E2 ) bundles as follows: E ⊕E2

(s1 ⊕ s2 ) := ∇1u s1 ⊕ ∇2u s2 ,

E ⊗E2

(s1 ⊗ s2 ) := (∇1u s1 ) ⊗ s2 + s1 ⊗ ∇2u s2 ,

∇u 1 ∇u 1

where u ∈ Vec(M) and s i ∈ Sec(Ei ), i = 1, 2. Notations for components. Let u ∈ Vec(M), s ∈ Sec(E) and g ∈ Sec(E∗ ). Then, ∇u s = u a ∇a s c e c ,

∇∗u g = u a ∇∗a g c e c ,

where ∇a s c := ∂ a s c + s b ω c ab ,

∇∗a g c := ∂ a g c − g b ω b ac .

Connection in operator-like form. If ∇ is a connection on E, then one can set ̃ := ∇a s c e c ⊗ dx a , ∇s on every coordinate domain. Thus, the mapping [131, 133] ̃ : Sec(E) → Sec(E ⊗ T ∗ M) , ∇ is well defined. Such a mapping satisfies the following properties:

(9.3)

212 | 9 Material Connections ̃ 1 ) Additivity: ∀s1 , s2 ∈ Sec(E) : ∇(s ̃ 1 + s2 ) = ∇s ̃ 1 + ∇s ̃ 2; ∇ ̃ ̃ ̃ 2 ) Homogeneity: ∀s ∈ Sec(E) ∀λ ∈ ℝ : ∇(λs) = λ∇s; ∇ ̃ 3 ) The Leibniz rule: ∀s ∈ Sec(E) ∀f ∈ C∞ (M) : ∇(fs) ̃ ̃ + s ⊗ df . ∇ = f ∇s ̃ generates the mapping ∇ via formula Conversely, given ∇ ̃ ⌞u . ∇u s := ∇s ̃ and simply write ∇. In the following, we omit the tilde symbol in ∇ Parallel transport. Let γ : 𝕀 → M be some smooth curve on M. A section s ∈ Sec(E) is said to be parallel along γ, if ∀t ∈ 𝕀 : ∇γ󸀠 (t) s = 0 . Choosing the local coordinate patch (U, x i ), one can express the above equation componentwise: ṡ a + ẋ c s b ω a cb = 0 , a = 1, . . . , rank(E) . Here, s = s a e a , γ󸀠 = ẋ a ∂ a and (⋅) ̇ denotes d/dt. Let p = γ(t0 ). Then one has the following initial condition: s p = s0 . Thus, locally, there exists a unique solution for the Cauchy problem ṡ a + ẋ c s b ω a cb = 0 , s ap

=

s0a

a = 1, . . . , rank(E) , ,

a = 1, . . . , rank(E) .

(9.4)

Therefore, one can define the parallel transport map P tt 0 : Eγ(t 0 ) → Eγ(t) as follows: with each element s0 ∈ Eγ(t 0 ) it assigns the value s(t) of the solution s for the Cauchy problem (9.4); P tt 0 : s0 󳨃→ s(t). Now, let u = γ󸀠 (t0 ) be the velocity vector of the curve γ at the point p and s ∈ Sec(E) be a smooth section. Then, the value ∇u s|p can be defined via the limit t

∇u s|p = lim

h→0

P t 00 +h s γ(t 0 +h) − s γ(t 0 ) h

=

󵄨󵄨 d t0 󵄨 P t s γ(t) 󵄨󵄨󵄨 . 󵄨󵄨t=t 0 dt

t

Here, P t 00 +h s γ(t 0 +h) belongs to Eγ(t 0 ) , and this means that for a sufficiently small h, the t

difference P t 00 +h s γ(t 0 +h) − s γ(t 0 ) is well defined and belongs to Eγ(t 0) . Thus, the element ∇u s|p ∈ Eγ(t 0 ) may be considered as a derivative of the section s at the point p along the vector u.

9.2 Affine Connection Let M be a smooth n-dimensional manifold. In the remainder of the chapter, we consider the case when the connection ∇ is given on the tangent bundle TM. That is, ∇ : Vec(M) × Vec(M) → Vec(M) .

9.2 Affine Connection | 213

In this case, we refer to ∇ as an affine connection, and coefficients ω a bc are denoted by Γ abc . Previous reasonings show that ∇ can be induced on T ∗ M and any other vector bundle that is constructed from copies of TM and T ∗ M via direct sum and tensor product operations. Such induced connections would be denoted similarly, by ∇. Expressions for covariant derivatives. If (∂ i ), (dx i ) are, respectively, the coordinate frame and the coframe, then from (9.1) and (9.2), it follows that ∇∂ a ∂ b = Γ cab ∂ c ,

∇∂ a dx b = −Γ bac dx c .

If T ∈ Sec(T (k,l) (TM)), T=T

i 1 ...i k

j1 ...j l ∂ i 1

⊗ ⋅ ⋅ ⋅ ⊗ ∂ i k ⊗ dx j1 ⊗ ⋅ ⋅ ⋅ ⊗ dx j l ,

then using (9.3), one obtains

∇i T

∇T = ∇i T

i 1 ...i k

= ∂i T

i 1 ...i k

−T

i 1 ...i k

i 1 ...i k j1 ...j l

j1 ...j l ∂ i 1

j1 ...j l

⊗ ⋅ ⋅ ⋅ ⊗ ∂ i k ⊗ dx j1 ⊗ ⋅ ⋅ ⋅ ⊗ dx j l ⊗ dx i ,

+T

b b...j l Γ ij1

b...i k

−

ik i1 i 1 ...b j1 ...j l Γ ib j1 ...j l Γ ib + ⋅ ⋅ ⋅ + T i ...i ⋅ ⋅ ⋅ − T 1 k j1 ...b Γ bij l .

9.2.1 The Transformation Law Transformation of connection coefficients. Let (e i )ni=1 and (ϑ i )ni=1 be the local frame and coframe defined, respectively, on an open set U ⊂ M, which is a domain for some chart. Assume that n2 smooth functions Ω i j : U → ℝ that define the smooth field [Ω i j ] : U → GL(n; ℝ) of invertible n × n matrices are given. This field generates the local frame n j U ∋ p 󳨃→ (e󸀠i |p )i=1 , e󸀠i = Ω i e j . i

The corresponding dual coframe U ∋ p 󳨃→ (ϑ󸀠 |p )ni=1 can be expressed through (ϑ i ) via 󸀠i

the relations ϑ = ℧i j ϑ j , where [℧i j ] = [Ω i j ]−1 . Denote the corresponding connection k

coefficients by Γ kij and Γ 󸀠 ij , that is, ∇e i e j = Γ kij e k ,

and ∇e󸀠i e󸀠j = Γ 󸀠

k

󸀠 ij e k

.

k

Obtain the transformation law from Γ kij to Γ 󸀠 ij . If (∂ x i ) is a coordinate frame defined j

on U, then e i = A i ∂ x j , where [A i j ] : U → GL(n; ℝ) is a uniquely determined smooth field of matrices. One obtains ∇e󸀠j e󸀠k = ∇(Ω sj e s ) (Ω k e q ) = Ω s j ∇e s (Ω k e q ) = Ω s j {Ω k ∇e s e q + e s (Ω k ) e q } . q

q

q

q

214 | 9 Material Connections Since Γ 󸀠

i

= ⟨ϑ󸀠 , ∇e j e󸀠k ⟩ = ℧i m ⟨ϑ m , ∇e󸀠j e󸀠k ⟩ and e s (Ω j ) = A l s ∂ x l Ω j , it follows that q

i

jk

Γ󸀠

i jk

= ℧i m Ω s j {Ω

q

q

q

k

⟨ϑ m , ∇e s e q ⟩ + ⟨ϑ m , e q ⟩ A l s ∂ x l Ω k } .

Taking relations Γ msq = ⟨ϑ m , ∇e s e q ⟩, and ⟨ϑ m , e q ⟩ = δ m q into account leads us to the final expression Γ󸀠

i jk

= Γ msq ℧i m Ω s j Ω

q k

+ A l s ℧i m Ω s j ∂ x l Ω m k .

(9.5)

The case of holonomic frames. The formula (9.5) determines the transformation law in the general case, when one changes the non-holonomic frame into a non-holonomic one. If the frame (e i ) is holonomic, that is, A l s = δ ls , then the relation (9.5) can be simplified: i q Γ 󸀠 jk = Γ msq ℧i m Ω s j Ω k + ℧i m Ω s j ∂ x s Ω m k . (9.6) If local frames (e i ) and (e󸀠i ) are both holonomic, then we are dealing with a change ∂x i i of coordinates. Thus, Ω i j = , where (x i ) generate (e i ) and (x󸀠 ) generate (e󸀠i ). The j 󸀠 ∂x expression (9.6) takes the form Γ󸀠

i

i jk

= Γ msq

i

∂x󸀠 ∂x s ∂x q ∂x󸀠 ∂2 x m + . ∂x m ∂x󸀠 j ∂x󸀠 k ∂x m ∂x󸀠 j ∂x󸀠 k

(9.7)

Formula (9.7) defines the connection transformation induced by the change of coordinates. Note that due to the last item in the right hand side of (9.7), such a transformation does not satisfy the transformation rule of the third-rank tensor field.

9.2.2 Torsion, Curvature, and Non-Metricity Let M be a smooth manifold with the Riemannian metric g and the connection ∇. The fact that the values of Γ ijk do not vanish is no evidence that the geometry on M is nonEuclidean. Indeed, it is easy to check that in Euclidean space Γ ijk = 0 only in Cartesian coordinates, while curvilinear coordinates make Γ ijk non-vanishing. The property of geometry, with which one associates the term “non-Euclidean”, is intimately related with tensor fields of torsion, T : Vec(M) × Vec(M) → Vec(M) , Riemann curvature, R : Vec(M) × Vec(M) × Vec(M) → Vec(M) , and non-metricity, Q : Vec(M) × Vec(M) × Vec(M) → C ∞ (M) .

9.2 Affine Connection | 215

These fields are defined via the following relations [15]: T(u, v) = ∇u v − ∇v u − [u, v] ; R(u, v, w) = ∇u ∇v w − ∇v ∇u w − ∇[u,v] w ; Q = −∇g .

(9.8) (9.9) (9.10)

Here, [⋅, ⋅] is the Lie bracket (13.20). Remark 9.1. The torsion tensor (9.8) is a measure of the connection symmetry. If (∂ i ) is a coordinate frame on M, then T(u, v) = ∇u v − ∇v u − [u, v] = u i (∂ i v k + v j Γ kij ) ∂ k − v j (∂ j u k + u i Γ kji ) ∂ k − (u i ∂ i v k − v j ∂ j u k ) ∂ k = u i v j (Γ kij − Γ kji ) ∂ k , from which follows the expression T = (Γ kij − Γ kji ) ∂ k ⊗ dx i ⊗ dx j . This means that T = 0 iff Γ kij = Γ kji , for all i, j, k. Remark 9.2. As follows from (9.10), the non-metricity tensor Q determines an incoherence of the connection ∇ with the Riemannian metric g. One can obtain the torsion T and the non-metricity Q via formulae (9.8) and (9.10) using a prescribed Riemannian metric (only non-metricity requires a metric) g and a connection ∇. Consider the opposite situation, when one has to define the affine connection ∇, corresponding to given Riemannian metric g, torsion T, and non-metricity Q. Denote by ∇ the desired connection. Choose the coordinate frame (∂ i ) and the coframe (dx i ). Taking into account (9.3), and deriving the expressions for Q and T in components, one can obtain the following system of 2n3 equations: Γ kij − Γ kji = T kij , ∂ i g jk − Γ

m

ij g mk

−Γ

m

ik g mj

= −Q ijk ,

i, j, k = 1, . . . , n , i, j, k = 1, . . . , n .

Cyclic permutations give the expressions −Q ijk = ∂ i g jk − Γ mij g mk − Γ mik g mj , −Q jki = ∂ j g ki − Γ mjk g mi − Γ mji g mk , −Q kij = ∂ k g ij − Γ mki g mj − Γ mkj g mi . Summation of the second and third equations and subtracting the first one from the result leads to the equality Q ijk − Q jki − Q kij = ∂ j g ik + ∂ k g ij − ∂ i g jk + T mij g mk + T mik g mj − Γ mjk g mi − Γ mkj g mi .

216 | 9 Material Connections

Thus, finally one has 2Γ mjk g mi = ∂ j g ik + ∂ k g ij − ∂ i g jk + T mij g mk + T mik g mj + T mjk g mi + Q jki + Q kij − Q ijk .

(9.11)

Since [g ij ] is invertible, the following relations for Γ ijk can be also derived: Γ ijk =

g im (∂ j g mk + ∂ k g mj − ∂ m g jk ) 2 g im + (T mjk + T kmj + T jmk ) 2 g im + (Q jkm + Q kmj − Q mjk ) , 2

(9.12)

where T kij = g km T mij . Formula (9.12) shows that it is sufficient to define the Riemannian metric and independent tensors Q and T to introduce an affine connection on the manifold M. Thus, (g, T, Q) can be considered as a triple of independent variables.

9.2.3 A Particular Case: Euclidean Space Let E be an affine Euclidean space with dimE = m. In the framework of conventional Euclidean geometry, a vector u is identified with the covector u♭ = g(u, ⋅), and vice versa. Here, g is a Riemannian metric that defines the scalar product (⋅). Accordingly, k m if (e k )m k=1 is a frame in E, then its corresponding dual is considered as a frame (e )k=1 s s s ♭ that satisfies the relations e k ⋅ e = δ k , s, k = 1, . . . , m. This means that ⟨(e ) , e k ⟩ = g(e s , e k ) = δ sk . Introduce the curvilinear coordinates² (q i ) in E. They are defined by functions q k = q k (x1 , . . . , x m ), which relate the Cartesian coordinates (x i ) with (q i ) and can be defined by m smooth diffeomorphisms. Frame and coframe fields are determined by the expressions ∂x s ∂q k ek = is k , ek = is s . ∂x ∂q Transition functions that match the curvilinear coordinates (q k ) with Cartesian ones, in fact, determine all geometrical quantities that correspond to connection and related fields. Indeed, it is easy to show that (i) The metric tensor g = g ij e i ⊗ e j and its corresponding dual, g ♯ = g ij e i ⊗ e j , which in conventional considerations is identified with g, have the following compo-

2 Recall that curvilinear coordinates were considered in Chapter 2, Section 2.2.

9.2 Affine Connection | 217

nents: g ij =

∂x k ∂x s δ ks , ∂q i ∂q j

g ij =

∂q i ∂q j ks δ . ∂x k ∂x s

(ii) The covariant derivative ∇ is defined by the relation ∇u v = u ⋅ ∇v ,

where

∇ := e k ∂ k ,

where ∂ k ≡ ∂q∂ k . (iii) Christoffel symbols are represented by the formula γ i jk = e i ⋅ ∇ej e k = e i ⋅

∂e k ∂q i ∂2 x p = . ∂x p ∂q j ∂q k ∂q j

Despite the fact that specific quantities like the metric and Christoffel symbols differ from accustomed Cartesian values, they do not correspond to a space more general than the Euclidean one. In order to show this, it is enough to calculate curvature, torsion, and non-metricity. To calculate the curvature note that i j k i j k i j ∇u ∇v w = u. .k .(∂ . . .k .v. .).(∂ . . .i w . . .). (∂ . . .j x) . . + u v (∂ k ∂ i w ) (∂ j x) + u v (∂ i w ) (∂ k ∂ j x) :::::::::::::::

k

i

j

k i

j

+ u (∂ k v ) w (∂ i ∂ j x) + u v (∂ k w ) (∂ i ∂ j x) + u k v i (∂ k ∂ i ∂ j x) , ::::::::::::::: :::::::::::::::

∇v ∇u w = v k (∂ k u i ) (∂ i w j ) (∂ j x) + v k u i (∂ k ∂ i w j ) (∂ j x) + v k u i (∂ i w j ) (∂ k ∂ j x) :::::::::::::::::

+ v k (∂ k u i ) w j (∂ i ∂ j x) + v k u i (∂ k w j ) (∂ i ∂ j x) + v k u i (∂ k ∂ i ∂ j x) , :::::::::::::::

:::::::::::::::

j k ∇[u,v] w = u. .i .(∂ u i (∂ i )w k (∂ j ∂ k x) . .i.v. .). (∂ . . .j w . . . ). .(∂ . .k. x) . . + ::::::::::::: :::::::::::::

− v i (∂ i u j ) (∂ j w k ) (∂ k x) − v i (∂ i u j ) w k (∂ j ∂ k x) . :::::::::::::::::

:::::::::::::::

Taking the symmetry of the second-order partial derivatives, ∂ i ∂ j x = ∂ j ∂ i x, into account and eliminating similar terms, one obtains ∀u, v, w ∈ Vec(E) :

R(u, v, w) = ∇u ∇v w − ∇v ∇u w − ∇[u,v] w ≡ 0 .

To calculate torsion one uses the following equalities: ∇u v = u i (∂ i v j ) (∂ j x) + u i v j (∂ i ∂ j x) , ∇v u = v i (∂ i u j ) (∂ j x) + v i u j (∂ i ∂ j x) , :::::::::::

[u, v] = u i (∂ i v j ) (∂ j x) − v i (∂ i u j ) (∂ j x) . :::::::::::

218 | 9 Material Connections

Thus, from the symmetry of the second derivatives, one obtains ∀u, v ∈ Vec(E) :

T(u, v) = ∇u v − ∇v u − [u, v] ≡ 0 .

Finally, calculate the non-metricity. Taking the relations g(∇u v, w) = w j u i (∂ i v j ) + w k u i v j (∂ k x) ⋅ (∂ i ∂ j x) , k i j g(v, ∇u w) = v j u i (∂ i w j ) + v:::::::::::::::: u w (∂ k x) ⋅ (∂ i ∂ j x) ,

u[g(v, w)] = u i w j (∂ i v j ) + u i v k (∂ i w k ) + u i v j w k (∂ i ∂ j x) ⋅ (∂ k x) i j k + u:::::::::::::::: v w (∂ j x) ⋅ (∂ i ∂ k x) ,

into account and using the symmetry of the second derivatives again, one arrives at ∀u, v, w ∈ Vec(E) :

Q(u, v, w) = g(∇u v, w) + g(v, ∇u w) − u[g(v, w)] ≡ 0 .

9.2.4 A Particular Case: Riemannian Space Riemannian space P is a Riemannian manifold together with a Levi–Civita connection on it. In Chapter 2, Section 2.4 such a connection was introduced from the requirement that autoparallel curves coincide with geodesics. It was shown in 2.4.4 that if g is a metric, then the Levi–Civita connection coefficients are given by γ k sq =

g kl (∂ q g ls + ∂ s g lq − ∂ l g sq ) . 2

In this case, the curvature R differs from zero generally, while torsion and nonmetricity vanish. To show that torsion is identically zero calculate the following expressions: ∇u v = u i (∂ i v j ) e j + u i v j γ k ij e k , ∇v u = v i (∂ i u j ) e j + v i u j γ k ij e k , [u, v] = u i (∂ i v j ) e j − v i (∂ i u j ) e j . Substitute them into (9.8) and taking into account the symmetry of the connection coefficients, γ k ij = γ k ji , cancel out similar items. As a result, we obtain the desired conclusion: ∀u, v ∈ Vec(P) : T(u, v) = ∇u v − ∇v u − [u, v] ≡ 0 .

9.2 Affine Connection | 219

In order to show that non-metricity also vanishes, we provide the following calculations: g(∇u v, w) = w k u i (∂ i v k ) + w k u i v j γ k ij , ::::::::

i

k

g(v, ∇u w) = v k u (∂ i w ) + v k u i w j γ k ij , ::::::::

i j k v w (∂ i g jk ) . u[g(v, w)] = u i w j (∂ i v j ) + u i v k (∂ i w k ) + u::::::::::

Since γ ijk + γ kji − ∂ j g ij = 0, we arrive at ∀u, v, w ∈ Vec(P) :

Q(u, v, w) = g(∇u v, w) + g(v, ∇u w) − u[g(v, w)] ≡ 0 .

Riemannian physical space as a model of the curvilinear substrate. Suppose that the physical space is represented by a three-dimensional affine-Euclidean space E. Let us consider shapes of a two-dimensional body B (material surface, in the terminology of [110]), which are placed on some fixed two-dimensional embedded submanifold S of E. Such a situation may hold when an LbL-process is considered: a twodimensional assembly is constructed from layers that are deposited on a substrate. In this situation, it may be convenient to consider the Riemannian space (S, g(s) ) ∗ g is the metric induced from E. Suppose as a physical space itself. Here, g(s) = ιS 1 2 3 that (x , x , x ) are Cartesian coordinates on E, and (u 1 , u 2 ) are local coordinates on S, which where obtained, for example, by applying the local k-slice condition (see Chapter 4, Section 4.5.2). Thus, the metric g(s) has the following dyadic decomposition (see (4.3)): g(s) = g αβ du α ⊗ du β ,

where

g αβ = δ ij

∂x i ∂x j . ∂u α ∂u β

The mapping x i : (u 1 , u 2 ) 󳨃→ x i (u 1 , u 2 ), i = 1, 2, 3, is the coordinate representation of the inclusion map ιS . 3D Euclidean description for S as for a surface. Since S is embedded into E, each point of S is described by the position vector field p : S → V; p(x) = x k (u 1 , u 2 )i k . Define the vector fields e α : S → V, α = 1, 2, as follows: eα =

∂p(u 1 , u 2 ) , ∂u α

α = 1, 2 .

For a point x ∈ S, the pair (e1 |x , e2 |x ) constitutes a basis of the vector space InS;x (T x S) ⊂ V. The space Tx S = InS;x (T x S) is the image of the tangent space T x S under the linear inclusion map (4.4) into the translation space V. Recall that in classical surface theory, the set x + Tx S (the subspace of the affine space E) is referred to as the tangent plane at x. Thus, (e α )2α=1 is the local frame on the surface S. In the framework

220 | 9 Material Connections

of the classical surface theory, the role of the Riemannian metric is played by the first fundamental form I: ∀u, v ∈ Tx S : I(u, v) := u ⋅ v . Its components are equal to I αβ = e α ⋅ e β and, thus, I αβ = g αβ . The first fundamental form I coincides with the Riemannian metric g(s) up to inclusion. At each point x ∈ S, the translation vector space V splits into the direct sum Tx S ⊕Nx S = V. In this expression, the item Nx S denotes the orthogonal complement of Tx S. It is a one-dimensional vector subspace of V with the basis n|x = ‖e 1 |x × e2 |x ‖−1 (e 1 |x × e 2 |x ). By the definition, the line x + Nx S is normal to the tangent plane x + Tx S. Thus, we obtain a unit normal vector field n : x 󳨃→ n|x on S. For every point x ∈ S, the principal part of the deviation of the surface from the tangent plane x + Tx S is given by the second fundamental form II x , which in classical notation is written as II|x = L x (du 1 )2 + 2M x du 1 du 2 + N x (du 2 )2 , L = n ⋅ ∂ u1 e 1 ,

M = n ⋅ ∂ u2 e 1 ,

N = n ⋅ ∂ u2 e 2 .

The Riemannian metric g(s) generates the Levi–Civita connection on S with the coefficients g αλ (∂ γ g λβ + ∂ β g λγ − ∂ λ g βγ ) , α, β, γ, λ = 1, 2 . 2 In this case, the curvature R, the Ricci tensor, and the scalar curvature are determined by (components are written in the coordinate frame) Γ αβγ =

R αβγδ = K(g αγ g δβ − g αδ g γβ ) , R αβ = Kg αβ , S = 2K , where K = det II/ det I is the Gaussian curvature of the surface S [134].

9.2.5 Connection on the Pullback Bundle Definition. Let M and N be smooth manifolds and 𝜘 : M → N be a C ∞ -diffeomorphism. Suppose that an affine connection ∇ is given on TN. Such a connection defines the connection ∇(𝜘) : Vec(M) × Sec(𝜘∗ TN) → Sec(𝜘∗ TN) , on the pullback bundle 𝜘∗ TN as follows: (𝜘)

∇u V := {∇𝜘∗ u (V ∘ 𝜘−1 )} ∘ 𝜘 .

(9.13)

Remark 9.3. Pushforward 𝜘∗ is a ℝ-linear mapping, and this implies (∇1 ) and (∇2 ). Next, if f ∈ C∞ (M), then 𝜘∗ u(f ∘ 𝜘−1 ) = u(f) and 𝜘∗ (fu) = f𝜘∗ (u). This implies (∇3 ) and (∇4 ).

9.2 Affine Connection | 221

Coordinate representations. Let (e i ) be smooth local frame for TN, then (E i ), where E i = e i ∘ 𝜘 is a local frame for 𝜘∗ TN. Let (x i ) be local coordinates on M and (y i ) be local coordinates on N. Denote the affine connection coefficients for ∇ by γ i jk . Then, one has (𝜘) ∇∂ xa E b = {∇𝜘∗ ∂ xa e b } ∘ 𝜘 . Denote T𝜘 = F i j ∂ y i ⊗ dx j . Since 𝜘∗ ∂ x a = (F i a ∘ 𝜘−1 )∂ y i , then (𝜘)

∇∂ xa E b = F i a (∇∂ yi e b ) ∘ 𝜘 . With the relation ∇∂ yi e b = γ c ib e c one finally obtains (𝜘)

∇∂ xa E b = F i a γ c ib E c . Thus, Γ cab = F i a γ c ib are connection coefficients for ∇(𝜘) . If u is a smooth section for TM and V is a smooth section for 𝜘∗ TN, then (𝜘)

∇u V = u a {∂ a V c + V b F i a γ c ib } E c . If (e i ) is a frame dual to (e i ), then the ordered tuple (E i ) of sections E i = e i ∘ 𝜘 is the corresponding dual frame for (E i ). Using Γ cab = F i a γ c ib and formula (9.2), one obtains (𝜘) ∇∂ xa E b = −F i a γ b id E d . The case of two-point tensors. We restrict ourselves to the case of the vector bundle 𝜘∗ TN ⊗ TM. Define the connection on it using (9.3). Let ∇(M) be an affine connection on M. Then (𝜘) (M) ∇u (s1 ⊗ s2 ) := (∇u s1 ) ⊗ s2 + s1 ⊗ ∇u s2 . ̃ α ) be a frame on M and let T : p 󳨃→ T iα (p)E i |p ⊗ E ̃ α |p be a two-point tensor. Let (E aκ β ̃ Then, ∇T = ∇β T E a ⊗ E κ ⊗ dx , in which ∇β T aκ = ∂ β T aκ + T aλ Γ κλβ + T bκ F c β γ a bc , and Γ κλβ are connection coefficients of ∇(M) . Application in continuum mechanics. Let B be a body, P be a physical space with the metric g and the Levi–Civita connection ∇, and 𝜘 be a configuration. As in Seĉ is a tion 9.2.4, the Levi–Civita connection can be induced to the shape S𝜘 . Since 𝜘 diffeomorphism, the above reasonings can be applied to the body B and the shape ̂ ∗ TS𝜘 can be endowed with the connection ∇(̂𝜘) . Note S𝜘 . Thus, the pullback bundle 𝜘 ∗ ̂ TS𝜘 can be regarded as tangent to the S𝜘 parts of the material that the elements of 𝜘 velocities V. Consider the particular case when dim B = dim P. The shape S𝜘 is an open subset of P. This means that the Levi–Civita connection on S𝜘 is ∇|S𝜘 . As we agreed previously, we omit the vertical bar and simply write ∇. Since TS𝜘 can be considered as

222 | 9 Material Connections ̂ ∗ TS𝜘 with 𝜘∗ TP. just a restriction of TP, i.e., TS𝜘 = ⨆x∈S𝜘 T x P, one can identify 𝜘 c (̂ 𝜘 ) ∗ i Then, ∇ is a connection on 𝜘 TP with coefficients Γ ab = F a γ c ib , where F i j are components of the configuration gradient F = F i j ∂ i ⊗ dXj .

9.2.6 The Moving Frame Method Let M be a smooth parallelizable manifold with dim M = n. From the parallelizability, it follows that there exists a global frame (z i )ni=1 for TM. This means that z i ∈ Vec(M) for all i = 1, . . . , n, and for each p ∈ M, the n-tuple (z i |p )ni=1 of the values is linearly independent. The moving frame method was proposed by E. Cartan in the case of n-dimensional affine-Euclidean space E and allows one to define an affine connection by a predefined field of linear transformations that act on the orthonormal frame (i k )nk=1 [135]. In our case, we have a smooth manifold with a non-trivial atlas, in general. To apply the idea of Cartans’ moving frame one needs: 1) to choose some admissible smooth atlas; 2) to construct an affine connection ∇|U on each charts’ domain U; 3) to “glue” these connections together. Thus, it is sufficient to restrict ourselves by considering a chart domain. Let U be some chart domain and (∂ i )ni=1 be the corresponding coordinate frame. This is a holonomic frame, i.e., [∂ i , ∂ j ] = 0 for all i, j. There exist such n2 smooth functions Ω i j : U → ℝ that form a smooth field [Ω i j ] : U → GL(n; ℝ), and ∀i ∈ {1, . . . , n} :

j

zi = Ω i ∂j .

The frame (z i )ni=1 is non-holonomic in general, that is, [z i , z j ] = −c ij k z k ,

(9.14)

where c ij k : U → ℝ are the objects of anholonomity. Using the relation (13.21) [fu, gv]p = f(p)g(p)[u, v]p + f(p)u p (g)v p − g(p)v p (f)u p , in which f, g are smooth functions defined on a neighborhood of p, one obtains the following system with unknowns c ij m in fixed i, j: q

q

−c ij m Ω k m = Ω i (∂ q Ω k j ) − Ω j (∂ q Ω k i ) ,

k = 1, . . . , n .

The solution for this system has the form q

q

c ij k = −℧k m {Ω i (∂ q Ω m j ) − Ω j (∂ q Ω m i )} , in which [℧i j ] = [Ω i j ]−1 . Taking the relation ∂ i ℧k j = −℧m j (∂ i Ω

q

k m) ℧ q

(9.15)

9.2 Affine Connection |

223

into account, one obtains a more convenient representation for the (9.15): q

c ij k = Ω m i Ω j (∂ m ℧k q − ∂ q ℧k m ) .

(9.16)

To the frame (z i )ni=1 there corresponds a coframe (ϑ i )ni=1 , which is defined by the relations ⟨ϑ i , z j ⟩ = δ ij , for all i, j. Herewith, ϑ i = ℧i j dx j . Let us obtain the coframe analog of (9.14). One has dϑ k = d(℧k j dx j ) = ∑ (∂ i ℧k j − ∂ j ℧k i ) dx i ∧ dx j = i R∘α;e .

In this formula, R∘i = R∘int and R∘α;e are the prescribed inner and outer radii of 𝜘∘α (B), ̃ α is a smooth function that represents the local parameter of deforrespectively, and b mation of Lα . Thus, (11.26) takes the form: R ∘α;e

R ∘ (α)

ext ̃ α (ξ) 1 ξ2 1 p e (α)−p i (α) ξ 2 −b , ] ξ dξ + ∫ [ ] ξ dξ = − ∫ [ 2− 2 2 ̃ 2 2 2 μ ξ ξ − b α (ξ) (ξ + A α ) (ξ + A α ) ∘ ∘

R int

R α;e

or, since the first integral may be calculated directly, R∘α;e √(R∘int ) + A α 2

ln

R∘int √(R∘α;e ) + A α 2

−

Aα 1 1 − ) ( 2 (R∘α;e )2 + A α (R∘ )2 + A α int

R ∘ext (α)

+ ∫ [ R ∘α;e

̃ α (ξ) 1 p e (α) − p i (α) ξ2 − b ] ξ dξ = . − ̃ α (ξ) (ξ 2 + A α )2 μ ξ2 − b

(11.28)

̃ α: Equation (11.27) should be rewritten in terms of b ̃ α (R∘ (α)) + A α = S(α) . b ext

(11.29)

Hence, for this case one, should solve the evolutionary problem (11.28) and (11.29) to ̃ α and A α . determine b

11.1.3 Results and Discussion To compare strain and stress distributions in discrete structurally inhomogeneous solids with similar distributions in solids with continuously distributed inhomogeneity consider following examples:

264 | 11 The Evolutionary Problem – Examples

a) A hollow cylinder assembled from a number of thin hollow cylindrical parts. We suppose that each part possesses a natural configuration. A sequence of assemblies with a common to all total volume V and an increasing number of parts (10, 30, 200, and 1000) is studied. b) A hollow cylinder with volume V, the same as in a), and piecewise continuous distribution for deformation parameter b α , which is taken to be the continuous approximation of the sequence of deformation parameters a s obtained for assembly from a) with a maximal numbers of parts. c) A hollow cylinder with volume V, the same as in a) and b), and piecewise continuous distribution for b α , which is obtained as the numerical solution of the evolutionary problem (11.28) and (11.29). From the general considerations discussed above, it is expected that the stress and strain distributions in solids with discrete inhomogeneity tend to the corresponding stress and strain distributions in a solid with some piecewise continuous b α , which can be obtained as a solution of the evolutionary problem (11.28) and (11.29) or can be represented as an approximation of the deformation parameters a s of sufficiently dense assembly. The following calculations illustrate this fact numerically. For calculations, we took the following values of parameters. For each of the assemblies, the inner radius ρ of the first cylindrical layer in reference configuration and its thickness ∆0 are taken equal to ρ = ∆0 = 2 mkm. The reference thicknesses ∆ k , k ≥ 1, for each of the assemblies are taken equal to⁵ ∆ k = 22/(N − 1) mkm, where N is the number of layers. The shrinkage coefficients are taken equal to S k = 0.7. Suppose that inner and outer hydrostatic loadings, p i,s and p e,s , vanish. In computations, we put β = 0.24. For assemblies, the recurrent equation systems (11.6) are solved. Strain and stress distributions are calculated by the equations obtained in Section 11.1.1. In Figures 11.2 and 11.3 the numerical results are shown, where the dotted line corresponds to ten layers, the dot-dashed line corresponds to 30 layers, and the dashed line corresponds to 200 layers. The case of 1000 layers is not shown on the graphs because its difference from continuous distributions discussed below is graphically indistinguishable. Angle brackets in Y denote that the corresponding component of Y is considered in a physical (with unit norm vectors) frame. The symbols Y , Y , and Y correspondingly denote radial, circumferential, and axial physical components of a tensor Y. So do Y , Y , Y , but relatively to the pair of cylindrical coordinate systems. Results for the continuously structural inhomogeneous solid that corresponds to the discrete cases considered above can be obtained in the following way. Let the mapping r 󳨃→ b α (r) given by b α (r) = −3.165, if r < 3.583 and b α (r) = −4.568 + 1.335 r +

5 These values were chosen according to [26].

11.1 Example: The Cylindrical Problem

(a)

(b)

(c)

(d)

(e)

(f)

|

265

Fig. 11.2: Relative physical components T /μ and P /μ of Cauchy and first Piola–Kirchhoff stress tensors: (a) radial Cauchy stress component, (b) radial first Piola–Kirchhoff stress component, (c) circumferential Cauchy stress component, (d) circumferential first Piola–Kirchhoff stress component, (e) axial Cauchy stress component, (f) axial first Piola–Kirchhoff stress component.

3.526 ⋅ 10−1 r2 + 8.298 ⋅ 10−3 r3 − 1.620 ⋅ 10−4 r4 , if r > 3.583, be the numerical approximation of the values for deformation parameters in the most dense fragmentation (1000 layers). The material metric in this case is Riemannian with non-trivial curvature. This fact is illustrated by the distribution of the Ricci invariant (11.16) shown in Figure 11.3 (f). Note that the Ricci invariant is equal to zero on the part of the material manifold corresponding to the initial body. Adhered material forms the part of the body with non-Euclidean material metrics. The corresponding Cauchy stress tensor T approx was obtained in Section 11.1.2, R∘ R∘ where for T α α α , the expression (11.23) is used with B∘α = 0. The corresponding first Piola–Kirchhoff stress tensor and the Eshelby energy-momentum tensor were also obtained in Section 11.1.2, where A α is to be equal to zero. These distributions are shown in Figures 11.2, 11.3 a, c, and e by solid lines.

266 | 11 The Evolutionary Problem – Examples

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 11.3: Relative physical components E ⟨ij⟩ /μ of the Eshelby energy-momentum tensor; distributions of the deformation parameter, and the first and Ricci invariants: (a) radial Eshelby energy momentum tensor component, (b) deformation parameter, (c) circumferential Eshelby energy momentum tensor component, (d) the first invariant of left Cauchy-Green tensor B, (e) axial Eshelby energy momentum tensor component, (f) Ricci invariant (scalar curvature).

The function

approx

x 󳨃→ δ(x) =

T rr

(x) − T discr rr (x)

max T discr rr

,

estimates the difference between stress states in solids with discrete and continuous inhomogeneities. A plot of this function is shown in Figure 11.4; label 1 designates 10 layers, label 2 designates 30 layers, label 3 designates 200 layers, and label 4 designates 1000 layers. In the right upper corner of Figure 11.4, the logarithms of absolute values for δ, i.e., ln |δ|, are shown.

11.2 Uniform Inflation of a Spherical Multilayered Structure

| 267

Fig. 11.4: Estimation of the difference between stress states in solids with discrete and continuous inhomogeneities.

11.2 Uniform Inflation of a Spherical Multilayered Structure 11.2.1 Layers and Assemblies Consider assemblies that consist of a finite number of spherical layers. We assume that for each layer, there exists a configuration which image, namely, the shape of the layer, is free of stresses. In the general case, these shapes are geometrically incompatible, i.e., they cannot be combined without gaps or overlaps. The totality of the reference shapes of the layers plays the same role as the natural shape of the body in the classical theory of elasticity; relative to these shapes, the strain measures, which are arguments of the function of the stored elastic energy, are calculated. Consider a family G = {Bα }α∈𝕀 of three-dimensional bodies, which are ordered with respect to the subset relation: ∀α, β ∈ 𝕀 : (α < β) ⇒ (Bα ⊂ Bβ ) . In addition, assume that Bα is a submanifold of Bβ . Speaking figuratively, this means that all bodies from G are topologically “uniform” and can be obtained from some “model” body. In our case, such a body is, for example, a spherical shell with unit exterior radius and with interior radius equal to 1/2. Consider the discrete case: 𝕀 = {0, 1, . . . , N}, N ≤ 1. Let the physical space be the Euclidean affine space E (Chapter 2, Section 2.1). Assume that the following conditions are satisfied: a) Actual shapes of bodies Bα ∈ G, as well as actual shapes of layers Lα , are spherical shells in E, i.e., they can be defined as the sets B α = {x ∈ E | r iα ≤ ‖x − o‖ ≤ r eα } , respectively.

L αβ = {x ∈ E | r iα,β ≤ ‖x − o‖ ≤ r eα,β } ,

268 | 11 The Evolutionary Problem – Examples b) Natural (stress-free) shapes of layers are represented by spherical shells in E, i.e., L α = {x ∈ E | R iα ≤ ‖x − o‖ ≤ R eα } . c)

Internal, R iα+1 , and external, R eα+1 , radii of stress-free shapes of Lα+1 are related with the external radius of the actual shape of the assembly Bα by relations R iα+1 = S α r eα ,

R eα+1 = R iα+1 + ∆ α+1 ,

where ∆ α , 1, 2, . . . , N are prescribed thicknesses of stress-free shapes of layers, and S α are prescribed quantities that represent gaps (if S α > 1) or overlaps (if S α < 1) of stress-free layers and characterize technological shrinkage due to the layer solidification. d) Internal, R0i , and external, R0e , reference radii for the initial body (the layer with number 0) are given. The following notation is used: a single lower index denotes the number of the corresponding layer or assembly, and the double lower index, whose elements are separated by a comma, are the assembly number and layer number in this assembly. Taking into account the accepted designations, the sequence of shapes of assemblies can be represented by the diagram 11.5. The assumptions a)–d) imply that for each assembly Bα , there exist configurations 𝜘α,β , β = 0, . . . , N, whose images are spherical shells in E. Parts of these shapes corresponding to layers with a number β, are free of stresses. In particular, for the assembly with number 3, there exist three such configurations; they are shown in the diagram 11.6. The area in dark color corresponds to a stress-free shape.

11.2.2 Coordinates and Vector Bases Since the physical space is Euclidean, one can choose the Cartesian coordinates (o, i 1 , i2 , i3 ) in it. Moreover, because all shapes of assemblies Bα and layers Lα are centrally symmetric, in addition to Cartesian coordinates (x1 , x2 , x3 ), it is convenient to use spherical coordinates (r, θ, φ) that are related with Cartesian via the formulae: x1 = r sin θ cos φ ,

x2 = r sin θ sin φ ,

x3 = r cos θ ,

where r > 0, θ ∈ ]0, π[, φ ∈ [0, 2π[ together with “deformed” spherical coordinates (r α,β , θ α,β , φ α,β ) that are related with (r, θ, φ) by the expressions r = ̃f α,β (r α,β ) ,

θ = θ α,β ,

φ = φ α,β .

(11.30)

11.2 Uniform Inflation of a Spherical Multilayered Structure

Fig. 11.5: Hollow ball assemblies.

Fig. 11.6: Family of shapes with stress-free layers.

| 269

270 | 11 The Evolutionary Problem – Examples Here, ̃f α,β : ℝ+ → ℝ+ is a homeomorphism onto its image that can be defined as the extension of a function f α,β : [R iβ , R eβ ] → [r iα,β , r eα,β ]. The function⁶ f α,β specifies the distance from the origin o to the actual positions of material points of the shape L α,β of the layer Lβ in the assembly Bα depending on their reference positions in the reference shape L β . Each of these coordinate systems is associated with the shape of the corresponding layer and plays the role of “frozen” (convected) coordinates. Thus, an extended atlas of physical space is constructed. In addition to the Cartesian chart, such an atlas contains charts that are defined by various variants of spherical coordinates. Such an extension makes it possible to simplify further calculations using different coordinates as needed. Each coordinate system generates a field of local bases, i.e., triples of vectors α,β α,β α,β (e r , e θ , e φ ) and (e r , e θ , e φ ), which are tangent to the corresponding coordinate lines and are defined by formulae ec =

∂x s is , ∂c

α,β

c ∈ {r, θ, φ} ;

ec =

∂x s i s , c α,β ∈ {r α,β , θ α,β , φ α,β } . ∂c α,β

Dual bases (e r , e θ , e φ ) and (e r , e θ , e φ ) are defined by the biorthogonality relations α,β

α,β

α,β

α,β

e s ⋅ e c = δ sc , s, c ∈ {r, θ, φ} ;

e s ⋅ e c = δ sc , s, c ∈ {r α,β , θ α,β , φ α,β } . α,β

The exact expressions for the bases are given in Tab. 11.2. Tab. 11.2: Bases for different coordinates. i1

i2

i3

sin θ cos φ r cos θ cos φ −r sin θ sin φ

sin θ sin φ r cos θ sin φ r sin θ cos φ

cos θ −r sin θ 0

eθ

r α,β cos θ cos φ

r α,β cos θ sin φ

−r α,β sin θ

eφ eθ eφ eθ

α,β

α,β

−r α,β sin θ sin φ r −1 cos θ cos φ −(r sin θ)−1 sin φ r −1 α,β cos θ cos φ

r α,β sin θ cos φ r −1 cos θ sin φ (r sin θ)−1 cos φ r −1 α,β cos θ sin φ

0 −r −1 sin θ 0 −r −1 α,β sin θ

eφ

−(r α,β sin θ)−1 sin φ

(r α,β sin θ)−1 cos φ

0

α,β

er = er = er = er α,β eθ eφ α,β

α,β

6 In the following, it is necessary to express the reference radius through the actual one and formulate the differential relations. In this regard, the function f α,β must be bijective and infinitely many times continuously differentiable together with its inverse.

11.2 Uniform Inflation of a Spherical Multilayered Structure

| 271

11.2.3 Strain Measures Since the deformation of each layer Lβ in the body Bα in the natural shape is centrally symmetric (11.30) (radial displacements are given by the function f α,β ), the deformation gradient, its transpose, and the Cauchy–Green strain tensors can be represented by diagonal dyadic expansions 󸀠 F α,β = f α,β er ⊗ er + eθ ⊗ eθ + eφ ⊗ eφ , α,β

F Tα,β B α,β B−1 α,β

α,β

α,β

f α,β 2 α,β f α,β 2 α,β 󸀠 = f α,β er ⊗ er + ( ) eθ ⊗ eθ + ( ) eφ ⊗ eφ , r α,β r α,β f α,β 2 f α,β 2 2 T 󸀠 = Fα,β Fα,β = (f α,β ) er ⊗ er + ( ) eθ ⊗ eθ + ( ) eφ ⊗ eφ , r α,β r α,β r α,β 2 r α,β 2 1 2 = ( 󸀠 ) er ⊗ er + ( ) eθ ⊗ eθ + ( ) eφ ⊗ eφ . f α,β f α,β f α,β α,β

(11.31) (11.32)

The principal invariants of tensors Bα,β are defined by the relations 󸀠 I1 (B α,β ) = tr B α,β = (f α,β ) + 2( 2

f α,β 2 ) , r α,β

2 f α,β 4 1 2 2 f α,β 󸀠 ) ( ) +( ) , (I1 (B α,β ) − I1 (B2α,β )) = 2(f α,β 2 r α,β r α,β 4 2 f α,β 󸀠 ) ( ) . I3 (B α,β ) = det B α,β = (f α,β r α,β

I2 (B α,β ) =

11.2.4 Incompressible Material We assume that the material of the layers is incompressible, homogeneous, and isotropic, and the density of the elastic energy stored by it during deformation is determined by the two-constant Mooney–Rivlin potential, W α,β = C1 (I1 (B α,β ) − 3) + C2 (I2 (B α,β ) − 3) .

(11.33)

The constants (elastic modules) C1 and C2 can be expressed through another pair of modules b and μ as follows: C1 = (1 + b)μ/4, C2 = (1 − b)μ/4. The use of the modules C1 and C2 later allows the expressions obtained to be written in a more concise form, while the μ and b modules indicate the relationship between the non-linear hyperelastic material model and the linear elastic model; for small deformations from the unstressed state, μ coincides with the shear modulus of a linearly elastic incompressible material. For the potential (11.33), the Rivlin formula gives the Cauchy stresses: T α,β = −p α,β I + 2C1 B α,β − 2C2 B−1 α,β ,

272 | 11 The Evolutionary Problem – Examples where I is the identity tensor, p α,β is the hydrostatic loading, which is determined from the equilibrium equation ∇ ⋅ T = 0 and the prescribed boundary conditions. For a centrally symmetric field, they can be represented in the form α,β

∂ T r r 2 α,β r α,β θ : + (T r − T θ) = 0 , ∀r ∈ ∂r r α,0 󵄨 α,α 󵄨 r 󵄨󵄨 i r 󵄨󵄨 T r 󵄨󵄨 i = q α , T r 󵄨󵄨 e = q eα , 󵄨r=r α,0 󵄨r=r α,α r 󵄨󵄨 r 󵄨󵄨 T α,β ⋅ e 󵄨󵄨r=r e = T α,β+1 ⋅ e 󵄨󵄨r=r i , r eα,β = r iα,β+1 . α,β α,β+1 ]r iα,β , r eα,β [

(11.34) (11.35) (11.36)

Boundary conditions (11.35) characterize the uniform hydrostatic loading of the assembly of spherical boundaries, while equalities (11.36) state the conditions of ideal contact between the layers. Solutions of the equation (11.34) can be obtained in the form 2/3

α,β

α,β

T rr = p 0 +

α,β

C1 (5r3 − a α,β ) (r3 − a α,β )

r4 (r3 − a α,β )

α,β

T θ θ = T rr + 2

α,β

T

a α,β (2r3 − a α,β ) r4

(r3

4/3

− a α,β )

+ 2C2 r2 (r3 + a α,β ) 1/3 2/3

(C1 (r3 − a α,β )

,

− C 2 r2 ) ,

(11.37)

α,β

φ φ

= T θθ ,

α,β

where p 0 , a α,β are constants, which must be chosen so that the boundary conditions (11.35) and (11.36) are satisfied. Substitution of (11.37) into them results in a system of non-linear algebraic equations. For the initial body (α = 0), the system consists 0,0 of two equations with two unknowns, p 0 , a0,0 : 0,0

p 0 + T [R0i ; a0,0 ] = q0i ,

0,0

p 0 + T [R0e ; a0,0 ] = q0e ,

(11.38)

where −1

T[R; a] = (R(R3 + a)4/3 )

{C1 (5R3 + 4a)R2 + 2C2 (R3 + a)2/3 (R3 + 2a)} .

Systems for other assemblies Bα , 0 < α ≤ N (the relations R eβ = R iβ + ∆ β , an 0 < β < α are taken into account) consist of 2α equations relative to 3α − 1 unknowns α,0

α,1

α,α

p 0 , p 0 , . . . , p 0 , a α,0 , a α,1 , . . . , a α,α , R1i , R2i , . . . , R iα : α,0

α,α

p 0 + T [R0i ; a α,0 ] = q iα ,

α,1

p 0 + T [R eα ; a α,α ] = q eα ,

α,2

p 0 + T [R1e ; a α,1 ] = p 0 + T [R2i ; a α,2 ] , 3

3

(R1e ) + a α,1 = (R2i ) + a α,2 , ... α,α−1

p

α,α

0

+ T [R eα−1 ; a α,α−1 ] = p 0 + T [R iα ; a α,α ] , 3

3

(R eα−1 ) + a α,α−1 = (R iα ) + a α,α .

(11.39)

| 273

11.2 Uniform Inflation of a Spherical Multilayered Structure

The condition c) gives N − 1 equations that complement the system (11.39): R iα = S α √(R eα−1 ) + a α−1,α−1 , 3

3

0 n is linearly dependent. The number n is called the dimension of vector space V and is denoted by dim V. By convention, dim{0} := 0. If for any n ∈ ℕ, one can find n + 1 linearly independent sets of vectors, then V is called infinite dimensional.

12.5 Linear Spaces and Mappings | 303

Let V be a vector space. An ordered n-tuple (e1 , . . . , e n ) of vectors from V is called a basis for V if the set {e1 , . . . , e n } ⊂ V is linearly independent, and for any vector v ∈ V, there exist real numbers λ i ∈ ℝ, i = 1, . . . , n, such that v = λ1 e 1 + ⋅ ⋅ ⋅ + λ n e n . From the linear independence property, it follows that such numbers λ i are unique. For a vector v ∈ V, the corresponding numbers λ i are called components and are denoted by v i . Using the Einstein summation convention, one can write v = v i e i . The number n of vectors that constitute a basis does not depend on the basis and is equal to the dimension of V: n = dim V. Any linearly independent set of n = dim V vectors forms a basis for V [42].

12.5.2 Linear and k-Linear Mappings Vector space of linear mappings. Linear mappings between vector spaces preserve the linear relations between arguments. More precisely, the definition of linear mappings can be formulated as follows. Let V, W be vector spaces. A mapping L : V → W is called a linear mapping (transformation) if L1 ) (additivity) ∀u, v ∈ V : L(u + v) = L(u) + L(v); L2 ) (homogeneity) ∀u ∈ V ∀λ ∈ ℝ : L(λu) = λL(u). The set of all linear mappings from V to W can be endowed with a vector space structure. To this end, define the linear mappings on V by the relations (L + M) : V → W ,

(L + M)(u) := L(u) + M(u) ,

(λL) : V → W ,

(λL)(u) := λL(u) ,

where L, M ∈ W V are linear mappings, and λ ∈ ℝ is a scalar. This turns the set of all linear mappings from V to W together with the operations defined above into a vector space denoted by Lin(V; W). Note that if dim V = n and dim W = m, then dim Lin(V; W) = mn [42]. If L ∈ Lin(V; W) is a bijective mapping, then its inverse, L −1 , is similarly linear; −1 L ∈ Lin(W; V). In this book, the set of all invertible linear mappings from V to W is denoted by Inv(V; W). Let V, U and W be vector spaces. If L ∈ Lin(V; U) and M ∈ Lin(U; W), then M ∘ L ∈ Lin(V; W). Often, the composition M ∘ L of linear mappings is simply denoted by ML. Isomorphism. Let V and W be vector spaces. A bijective mapping L ∈ Lin(V; W) is called an isomorphism. If for V and W at least one isomorphism exists, then these vector spaces are called isomorphic. In this case, the following designation is used:

304 | 12 Algebraic Structures V ≅ W. For finite dimensional spaces V and W, the following necessary and sufficient condition holds: V ≅ W iff dim V = dim W [42]. In particular, for any ndimensional vector space V, the following isomorphism Φ : V → ℝn exists. Indeed, let (e i )ni=1 be a basis for V. Then define Φ as Φ : V → ℝn ,

Φ(v) := (v1 , . . . , v n ) ,

for

v = vi ei .

Such a mapping is an isomorphism, and one has that V ≅ ℝn . This isomorphism depends on a particular choice of basis. Any isomorphism transforms a basis into another basis. Let V, W be finite dimensional isomorphic vector spaces. Suppose that dim V = n. If L ∈ Lin(V; W) is the corresponding isomorphism, and (e i )ni=1 is a basis for V, then (L(e i ))ni=1 is a basis for W. Thus, for any vector w ∈ W one has the unique decomposition w = w i L(e i ). Dual vector space. Suppose that V is a vector space. Any linear transformation ν ∈ Lin(V; ℝ) is called a linear functional (covector) on V. The set of all linear functionals on V is denoted by V ∗ , i.e., V ∗ := Lin(V; ℝ). The vector space V ∗ is called the dual space of V. The action of a linear functional ν ∈ V ∗ on a vector v ∈ V is also denoted by angular brackets ⟨ν, v⟩. Here, ⟨⋅, ⋅⟩ : V ∗ × V → ℝ ,

⟨ν, v⟩ := ν(v) .

Let V be an n-dimensional vector space and (e i )ni=1 be some basis for V. Consider the n-tuple (e i )ni=1 of covectors e i ∈ V ∗ , i = 1, . . . , n, such that ⟨e i , e j ⟩ = δ ij ,

i, j = 1, . . . , n .

The n-tuple (e i )ni=1 forms a basis for V ∗ and is called the basis for the dual space V ∗ , dual with respect to the basis for V, or, for short, the dual (covector) basis to (e i )ni=1 [42]. It follows that dim V ∗ = n. If v ∈ V and v = v i e i , then v i = ⟨e i , v⟩ ,

i = 1, . . . , n .

These equalities mean that the dual basis represents the machinery that reads off the components of any vector v from V. Consequently, the components for a covector ν ∈ V ∗ can be expressed as expansion over (e i )ni=1 as follows (ν = ν i e i ): ν i = ⟨ν, e i ⟩ ,

i = 1, . . . , n .

Let V be an n-dimensional vector space. Since dim V ∗ = n, one has that V ≅ V ∗ . The corresponding isomorphism depends on a particular choice of basis. Reflexivity. The dual to the dual space is denoted by V ∗∗ := (V ∗ )∗ . There is a natural isomorphism between vector spaces V and V ∗∗ . This isomorphism is represented by canonical map ε : V → V ∗∗ , ε(v) = ε v ,

12.5 Linear Spaces and Mappings |

305

where ε v : V ∗ → ℝ is the mapping that acts on a covector ν as follows: ε v (ν) := ν(v). By the definition, ε v ∈ V ∗∗ and ε is an isomorphism [42] that does not depend on the choice of basis. In this regard, a vector v is often identified with the object ε v , and one writes v(ν) instead of ε v (ν). This property is referred to as reflexivity. Note that the infinite dimensional spaces may not be reflexible. k-linear mappings. The generalization of a linear mapping on the functions of several variables is given by the notion of k-linear mapping. Let V1 , . . . , Vk and W be vector spaces and L ∈ W V1 ×⋅⋅⋅×Vk be a mapping. Fix k vectors v i ∈ Vi , i = 1, . . . , k and define partial mappings as follows: L(v1 , . . . , v i−1 , ⋅, v i+1 , . . . , v k ) : Vi → W , L(v1 , . . . , v i−1 , ⋅, v i+1 , . . . , v k ) : x 󳨃→ L(v1 , . . . , v i−1 , x, v i+1 , . . . , v k ) . A mapping L ∈ W V1 ×⋅⋅⋅×Vk is called a k-linear mapping if for all vectors v i ∈ Vi , i = 1, . . . , k, and for any i = 1, . . . , k, the partial mapping L(v1 , . . . , v i−1 , ⋅, v i+1 , . . . , v k ) : Vi → W is linear. The set of all k-linear mappings from V1 × ⋅ ⋅ ⋅ × Vk to W can be endowed with vector space structure in a similar manner as was done for linear mappings. The resulting vector space is denoted by Link (V1 , . . . , Vk ; W). Algebra. An algebra over ℝ is a real vector space V endowed with a bilinear (twolinear) map L : V × V → V [42]. This map is called a product map, and its image on elements is often written as a juxtaposition. That is, the set V is endowed with three operations: +: V × V → V , +(u, v) = u + v , ⋅: ℝ × V → V , L: V × V → V ,

⋅(λ, v) = λ ⋅ u , L(u, v) = uv .

If L satisfies the property ∀u, v ∈ V : uv = vu , then the algebra V is called commutative. If the property ∀u, v, w ∈ V : (uv)w = u(vw) , holds for L, then the algebra V is called associative.

12.5.3 Tensor Products of Vector Spaces The ideas of the abstract tensor product are briefly introduced in the present section. We follow [1, 3, 145]. Apparently, for the first time, such a construction appeared in the treatise of N. Bourbaki [144].

306 | 12 Algebraic Structures Quotient vector space. Let U be a subspace of a vector space V. Consider the relation ∼U between vectors from V [42]: ∀u, v ∈ V : (u ∼U v) ⇔ (u − v ∈ U) . It is easy to show that this relation is an equivalence relation on V. Let x ∈ V and [x] be the corresponding equivalence class. It can be shown that [x] = {x + u | u ∈ U}. Indeed, [x] = {v ∈ V | v ∼U x} = {v ∈ V | v − x ∈ U} = {v ∈ V | ∃u ∈ U : v = x + u} = {x + u | u ∈ U} . The quotient set of V with respect to ∼U is denoted by V/U. It can be endowed with a real vector space structure as follows. Let a ∈ V/U be an equivalence class. If λ ∈ ℝ and u, v ∈ a, then λu − λv ∈ U and λu ∼U λv. Taking this into account, one can define λa as λa := [λu] , for some u ∈ a. Thus, the scalar multiplication ⋅ : ℝ × V/U → V/U ,

(λ, a) 󳨃→ λa ,

is defined. Let a, b ∈ V/U be an equivalence classes. Take representatives x1 , x2 ∈ a, y1 , y2 ∈ b. One can obtain that x1 + y1 ∼U x2 + y2 . Thus, the definition a + b := [x + y] , where x ∈ a and y ∈ b, is correct. The addition operation is defined as follows: + : V/U × V/U → V/U ,

(a, b) 󳨃→ a + b .

With addition and scalar multiplication operations, the set V/U becomes a vector space. This is called the quotient space of V modulo U. Formal linear combinations. The notion of the tensor product is based on the idea of a formal linear combination. Such an idea allows one to define finite sums like ∑m i=1 a i x i , where a i are real numbers, and x i are elements of a set X. Suppose that X is a set. A formal linear combination of elements of X [3] is a scalarvalued function f : X → ℝ, such that the set K f := {x ∈ X | f(x) ≠ 0} = f −1 (ℝ \ {0}) of all elements x ∈ X that do not transform to zero by f is a finite set. The set M(X) of all such functions is a real vector space with the typical operations of pointwise addition and scalar multiplication. Its basis is formed by the functions (δ x )x∈X from M(X), which can be defined as follows: {1, δ x (y) := { 0, {

y=x, y ≠ x .

12.5 Linear Spaces and Mappings | 307

Each element f ∈ M(X) has the unique decomposition f = ∑ki=1 a i δ x i , where x1 , . . . , x k are the elements of X, for which f(x) ≠ 0 and a i = f(x i ). By the definition of K f , the sum is finite. Typically, the function δ x is identified with x, and the set X is thought of as the subset of M(X). Thus, one writes f = ∑ki=1 a i x i . Tensor product. Suppose that V1 and V2 are real vector spaces. The set V1 ×V2 consists of all possible pairs (v1 , v2 ) and v i ∈ Vi , and can be endowed with a vector space structure by coordinatewise addition and coordinatewise scalar multiplication. Then, the expressions (where v i , v󸀠i , v󸀠󸀠i ∈ Vi , λ ∈ ℝ) 󸀠 󸀠󸀠 (v󸀠1 + v󸀠󸀠 1 , v 2 ) − (v 1 , v 2 ) − (v 1 , v 2 ) ,

󸀠 󸀠󸀠 (v1 , v󸀠2 + v󸀠󸀠 2 ) − (v 1 , v 2 ) − (v 1 , v 2 ) ,

λ(v1 , v2 ) − (λv1 , v2 ) ,

λ(v1 , v2 ) − (v1 , λv2 ) ,

(12.2)

are not equal to zero. Consider the space M(V1 × V2 ) of all formal linear combinations. Let N be its subspace that spans over all possible differences (12.2). The tensor product V1 ⊗ V2 [144] is the quotient vector space V1 ⊗ V2 := M(V1 × V2 )/N . The equivalence class of an element (v1 , v2 ) is denoted by v1 ⊗ v2 := [(v1 , v2 )] , and is called the tensor product of v1 and v2 [3]. The tensor product ⊗ satisfies 󸀠 󸀠󸀠 (v󸀠1 + v󸀠󸀠 1 ) ⊗ v2 = v1 ⊗ v2 + v1 ⊗ v2 ,

λ(v1 ⊗ v2 ) = (λv1 ) ⊗ v2 ,

󸀠 󸀠󸀠 v1 ⊗ (v󸀠2 + v󸀠󸀠 2 ) = v1 ⊗ v2 + v1 ⊗ v2 ,

λ(v1 ⊗ v2 ) = v1 ⊗ (λv2 ) .

Tensor product and associativity. Let U, V, W be three vector spaces. Since (U⊗V)⊗ W is naturally isomorphic to U ⊗ (V ⊗ W), one can simply write U ⊗ V ⊗ W for these sets and u ⊗ v ⊗ w for their elements. Thus, one inductively comes to the products V1 ⊗ ⋅ ⋅ ⋅ ⊗ Vk and v1 ⊗ ⋅ ⋅ ⋅ ⊗ v k for k vector spaces and their elements. Basis and dimension of tensor product. Suppose that V1 and V2 are real vector spaces (i) n i are bases for Vi , i = 1, 2, then the collection with dimensions n1 and n2 . If (e j )j=1 (1)

(2)

(e i1 ⊗ e i2 )

1≤i 1 ≤n 1 ,1≤i 2 ≤n 2

,

is the basis for V1 ⊗ V2 . Thus, dim(V1 ⊗ V2 ) = n1 n2 [3]. Tensor product and linear mappings. Suppose that V1 , . . . , Vk are finite-dimensional vector spaces. It can be proven that the following natural isomorphisms exist: V1 ⊗ ⋅ ⋅ ⋅ ⊗ Vk ≅ Link (V1∗ , . . . , Vk∗ ; ℝ) , V1∗ ⊗ ⋅ ⋅ ⋅ ⊗ Vk∗ ≅ Link (V1 , . . . , Vk ; ℝ) . That is, the tensor product of k vector spaces is naturally isomorphic to the vector space of k-linear mappings.

308 | 12 Algebraic Structures

12.5.4 Vectors and Linear Mappings in Euclidean Space Inner (scalar) product. The vector space structure does not presuppose the presence of a “measuring tool” that allows one to measure the lengths of vectors and angles between them. Such a tool is given by an additional structure: the Euclidean space structure. This structure is generated by the inner product. Let V be a vector space. An inner (scalar) product on V is a mapping (⋅) : V ×V → ℝ with the following properties: (i) Symmetry. ∀u, v ∈ V : u ⋅ v = v ⋅ u. (ii) Linearity in the first coordinate. ∀v1 , v2 , w ∈ V ∀λ1 , λ2 ∈ ℝ : (λ1 v1 + λ2 v2 ) ⋅ w = λ1 (v1 ⋅ w) + λ2 (v2 ⋅ w) . (iii) Positive definiteness. ∀v ∈ V : v ⋅ v ≥ 0. Moreover, ∀v ∈ V : (v ⋅ v = 0 ⇔ v = 0) . In this case, (V, ⋅) is called a Euclidean vector space, vectors from V are denoted by Latin boldface minuscules, and linear mappings between Euclidean vector spaces are denoted by Latin boldface majuscules. That is, v ∈ V, L ∈ Lin(V; W). Such notations are used in classical works in mechanics [37, 47, 59, 104, 110, 146] and seem to be a tribute to the notation. Dual vector basis. Suppose that (V, ⋅) is an n-dimensional Euclidean vector space. Let (e i )ni=1 be a basis for V. In conventional mechanics, one uses the dual vector basis (e i )ni=1 [37]. This basis is defined by the relations e i ⋅ e j = δ ij ,

i, j = 1, . . . , n .

Denote by g ij = e i ⋅ e j , i, j = 1, . . . , n, the components of the bilinear form that defines the inner product (⋅). Note that g ij are also called the components of the metric tensor, but we reserve this term for the corresponding field (see Chapter 2, Section 2.2.5). The n × n matrix [g ij ] is symmetric, non-degenerate, and such that g = det[g ij ] > 0. The elements of the dual basis (e i )ni=1 can be decomposed over the basis (e i )ni=1 : e i = g ij e j , i = 1, . . . , n , where [g ij ] is the inverse matrix: g ik g kj = δ ij . A vector v ∈ V can be decomposed either over the basis (e i )ni=1 or over the dual basis (e i )ni=1 . In doing so, v = v i e i = v i e i , where v i = v ⋅ e i are contravariant components of v, while v i = v ⋅ e i are covariant components of v, i = 1, . . . , n. Therefore, the action of elements of the dual vector basis by scalar multiplication is similar to the direct action of the dual covector basis: both read off the components of a vector. Note that covariant and contravariant components of a vector v are related to each other via contractions with matrix [g ij ] or its

12.5 Linear Spaces and Mappings | 309

inverse: v i = g ij v j

and

v i = g ij v j ,

i = 1, . . . , n .

It can be proven that in n-dimensional Euclidean vector space (V, ⋅), there exists an orthonormal basis (i k )nk=1 , i k ⋅ i s = δ ks (it can be obtained by Gram–Schmidt orthogonalization [42]). The corresponding dual vector basis coincides with (i k )nk=1 , but formally we will use the upper index notation (i k )nk=1 even in this case, for consistency. Theorem (The Riesz representation theorem). Let (V, ⋅) be a finite dimensional Euclidean vector space. For any linear functional ν ∈ V ∗ , there exists a unique vector w ∈ V for which ∀v ∈ V : ν(v) = v ⋅ w . Due to the importance of this theorem, we list one of the possible proofs below. Proof. 1. Uniqueness. Let ν ∈ V ∗ be a covector. Suppose that there exist at least two vectors w 1 , w 2 ∈ V, such that ν(v) = v ⋅ w 1 = v ⋅ w2 , for any vector v ∈ V. Using the linearity property of the inner product, one obtains that v ⋅ (w 1 − w 2 ) = 0 for an arbitrary vector v ∈ V. In particular, this equality holds for v = w 1 − w 2 , and the positive definiteness property implies that w 1 = w 2 . The uniqueness is proven. 2. Existence. The existence can be proven in the following manner. Suppose that dim V = n. Choose a basis (e k )nk=1 for V and the corresponding dual bases (e k )nk=1 for V ∗ and (e k )nk=1 for V. For a covector ν ∈ V ∗ , one has ν = ν k e k . If v ∈ V, then ν(v) = v k ν k , where v = v k e k . Set w := ν k e k . Then, v ⋅ w = v k ν k , and w is exactly the appropriate vector. Musical isomorphisms. The Riesz theorem allows us to establish mutually inverse correspondences between vectors and covectors. Indeed, if w ∈ V, then one can define a linear functional ν w ∈ V ∗ , such that for all v ∈ V, we have ν w (v) := v ⋅ w. The mapping (⋅)♭ : V → V ∗ , defined as v 󳨃→ v♭ := ν v , is an isomorphism. It is linear by construction. The injectivity is proven similarly as in the first part of the proof of the Riesz theorem, and the surjectivity follows directly from the Riesz theorem. Denote by w ν the vector dual (by the Riesz theorem) to a covector ν ∈ V ∗ . That is, for all v ∈ V, we have ν(v) := v ⋅ w ν . The mapping (⋅)♯ : V ∗ → V, which is defined as ν 󳨃→ ν♯ := w ν , is similarly an isomorphism. For this, it is sufficient to see that (⋅)♯ and (⋅)♭ are mutually inverse. Mutually inverse isomorphisms (⋅)♭ : V → V ∗

and (⋅)♯ : V ∗ → V ,

are called musical isomorphisms [3]. They establish the relations between the bilinear mapping ⟨⋅, ⋅⟩ and the inner product (⋅): ∀v, w ∈ V : ⟨v♭ , w⟩ = v ⋅ w , ∀w ∈ V

ν ∈ V ∗ : ν♯ ⋅ w = ⟨ν, w⟩ .

310 | 12 Algebraic Structures Consequently, if (e k )nk=1 is a covector basis, dual to (e k )nk=1 , then e i = (e i )♯

and

e i = (e i )♭ ,

i = 1, . . . , n .

Transposition. Let (V, ⋅V ) and (W, ⋅W ) be Euclidean spaces with scalar products ⋅V and ⋅W , respectively. For a linear map L ∈ Lin(V; W), there exists a unique map LT ∈ Lin(W; V) such that ∀u ∈ V

∀v ∈ W : v ⋅W L[u] = u ⋅V L T [v] .

(12.3)

This map is called the transpose of L. If L ∈ Lin(V; V) is a linear map with the property LT = L, then it is called symmetric. The set of all symmetric linear maps from V to V is denoted by Sym(V). Inner product in the space of linear mappings. The inner (scalar) product on V induces the inner product (:) : Lin(V; V) × Lin(V; V) → ℝ , on the space Lin(V; V) of linear mappings in the following way: ∀L, M ∈ Lin(V; V) : L:M = tr(L T M) .

(12.4)

It is easy to check that the so-defined mapping (:) satisfies all axioms for the inner product. Applying the Riesz theorem, we obtain the representation of a linear functional ε ∈ Lin(V; V)∗ , i.e., ε : Lin(V; V) → ℝ: ∀L ∈ Lin(V; V) : ε(L) = T ε :L , where T ε ∈ Lin(V; V) is a uniquely determined element of Lin(V; V). Orthogonal operator. Consider the special type of linear transformation, an orthogonal operator. The linear transformation Q ∈ Lin(V; V) is called the orthogonal operator if it preserves the inner product: ∀u, v ∈ V : Q[u] ⋅ Q[v] = u ⋅ v .

(12.5)

By definition, an orthogonal operator maps orthonormal bases to orthonormal bases. It is an invertible operator, and QQ T = I . Conversely, any linear operator with this property is orthogonal [42, 82]. Positive definiteness and square root. A linear map L is called positive definite if ∀u ∈ V : (u ≠ 0) ⇒ u ⋅ L[u] > 0 . For a positive definite operator L, the notion of square root is defined as follows [82]: there exists an operator M ∈ Lin(V; V) such that M 2 = L. This operator is unique and is denoted by √L.

12.6 Linear Groups

| 311

Norm. Let V be a vector space. A norm on V is a function ‖ ⋅ ‖ : V → ℝ with the properties: (i) Non-negativeness. ∀v ∈ V : ‖v‖ ≥ 0. (ii) Definiteness. If ‖v‖ = 0 then v = 0. (iii) Homogeneity. ∀v ∈ V ∀λ ∈ ℝ : ‖λv‖ = |λ|‖v‖. (iv) Triangle inequality. ∀v, w ∈ V : ‖v + w‖ ≤ ‖v‖ + ‖w‖. Suppose that (V, ⋅) is a Euclidean vector space. The scalar products (⋅) and (:) on V and Lin(V; V) generate norms on these spaces as follows: ‖ ⋅ ‖V : V → ℝ , ‖ ⋅ ‖Lin(V;V) : Lin(V; V) → ℝ ,

‖u‖2V := u ⋅ u , ‖L‖2Lin(V;V) := L:L .

12.6 Linear Groups The general linear groups Let s ∈ ℕ. Consider the set of all invertible s × s matrices with real elements. This set, together with the ordinary matrix multiplication operation, forms a group denoted by GL(s; ℝ). This group is called the general linear group of degree s. The automorphism group. Let V be an s-dimensional real vector space. The set of all possible isomorphisms from V onto V (automorphisms) together with the composition defines a group GL(V). This group is called the automorphism group. Isomorphism GL(s; ℝ) ≅ GL(V). If two groups are isomorphic, then one of them may be treated as a representation of the other. Consider the groups GL(V) and GL(s; ℝ), where V is an s-dimensional real vector space. These groups are isomorphic. Indeed, if one fixes some basis (e i )si=1 for V, then for every automorphism L ∈ GL(V), one has L(e j ) = Ω i j e i , i, j = 1, . . . , s . The correspondence L 󳨃→ [Ω i j ] is the desired isomorphism. Thus, the group GL(s; ℝ) can be considered as a “coordinate” representation of the group GL(V). The orthogonal group. Let V be an s-dimensional Euclidean vector space. The orthogonal operators T ∈ Lin(V; V) constitute an orthogonal group under the composition operation. This group is denoted by O(s) and is isomorphic to the group of s × s orthogonal matrices².

2 A square matrix Q is called orthogonal if QQ T = Q T Q = I.

312 | 12 Algebraic Structures

12.7 Affine Space Affine space (over ℝ) is a triple (A, V, ψ) such that the following axioms are satisfied: A1 ) A is a non-empty set, whose elements are referred to as points. A2 ) V is a vector space over ℝ, whose elements are referred to as translation vectors. A3 ) ψ : A × A → V is a mapping, which assigns to each ordered pair (a, b) ∈ A × A some vector from V, denoted by an arrow over concatenated symbols of points, 󳨀→ ab, or by formal subtraction, b − a. The mapping ψ satisfies the following Weyl axioms [35]: W1 ) for every points a, b, c ∈ A the following Chasles’ relation holds: 󳨀→ 󳨀 → 󳨀 → ab + bc + ca = 0 ∈ V . W2 ) for any point a ∈ A and for any vector v ∈ V, there exists a unique point 󳨀→ b ∈ A, such that ab = v. Consider the elementary consequences from the definition. Taking a = b = c in the 󳨀→ Weyl axiom W1 ) implies that aa = 0. If one takes in W1 ) a = c, then it leads to the 󳨀→ 󳨀→ relation ab = −ba. Using the latter equality, the Chasles relation can be rewritten in the form 󳨀→ 󳨀 → 󳨀 → ab + bc = ac . According to the Weyl axiom W2 ), for any fixed point a ∈ A, the mapping ψa : A → V ,

󳨀→ ψ a (x) := ax ,

is a bijection. This allows one to define the external operation +: A × V → A ,

+ : (a, v) 󳨃→ a + v := ψ−1 a (v) ,

󳨀→ which assigns to each ordered pair (a, v) a unique point b ∈ A, such that ab = v. From the definition of (+), it follows that (+1 ) if a, b ∈ A are two arbitrary points, then there exists a unique vector u ∈ V, such that a + u = b; (+2 ) for any point a ∈ A, the following identity holds: a + 0 = a; (+3 ) for any point a ∈ A and all vectors u, v ∈ V, one has (a + u) + v = a + (u + v). Remark 12.1. Properties (+1 )–(+3 ) of the operation + are often stated as affine space axioms [144]. In this case, the affine space is defined as the ordered triple (A, V, +), where A is a non-empty set, V is a real vector space, and +: A × V → A ,

+ : (a, v) 󳨃→ a + v ,

is an external binary operation that satisfies the conditions (+1 )–(+3 ).

12.7 Affine Space |

313

󳨀→ Let ab denote such a unique vector u ∈ V, that a + u = b. For the mapping 󳨀→ A × A → V , (a, b) 󳨃→ ab , the Weyl axioms W1 ) and W2 ) follow from these conditions. The particular case of affine space is represented by the triple (V, V, ψ), where V is a real vector space and ψ : V × V → V is a mapping such that ψ : (a, b) 󳨃→ b − a. In this case, A = V, and vectors from V are “points” of the affine space. The dimension of affine space (A, V, ψ) is the dimension of the space V: dim A := dim V. In this book, we restrict ourselves to finite dimensional affine spaces. In the affine space (A, V, ψ), dim A = n, one can naturally introduce the notion of coordinates. Let (e i )ni=1 be a basis for V and let o ∈ A be a fixed point (origin). The ordered pair (o, (e i )ni=1 ) is called an affine coordinate system. If x ∈ A is arbitrary 󳨀 → point, then its coordinates with respect to (o, (e i )ni=1 ) are components of the vector ox n with respect to the basis (e i )i=1 . The properties of the mapping ψ imply that any point x ∈ A is uniquely characterized by its coordinates (x i )ni=1 and x = o + xi ei . Suppose that a, b ∈ A are two points that are represented in (o, (e i )ni=1 ) by coordinates 󳨀→ 󳨀→ 󳨀→ 󳨀→ (a i )ni=1 and (b i )ni=1 , respectively. Then, ab = ao + ob = (b i − a i )e i , and the vector ab has the components (b i − a i )ni=1 . The previous reasonings show that the affine space reflects the classical notions of analytical geometry. In particular, it supports the ideas of a vector and a coordinate. If one chooses a coordinate system, then it can consider points as ordered tuples. The difference between tuples gives the components of a vector that connects the corresponding points. Affine space results in all possible translations from a fixed origin. Let (A, V, ψ) be an affine space with dim A = n. Suppose that U ⊂ V is a subspace of V and a ∈ A is some point. Then, the set F := a + U = {a + h | h ∈ U} , is called the affine subspace of dimension dim U [147]. If dim F = 1, then F is called an affine line. In this case, the subspace U is generated by a non-zero vector h ∈ U, and one can write F = {a + λh | λ ∈ ℝ} . An affine subspace of dimension 2 is called an affine plane. If dim U = n − 1, then the affine subspace is called an affine hyperplane. Two affine subspaces F1 = a1 +U1 and F2 = a2 +U2 are called strongly parallel, if U1 = U2 . They are called weakly parallel if U1 ⫋ U2 . Thus, affine space is an absolute world of lines and planes; the world that is represented in classical synthetic geometry. At the same time, the definition of affine space does not presuppose any metric data. In particular, the notion of an angle is undefined. To build the analog of synthetic Euclidean space, one needs an additional tool. In Chapter 2, such tool is provided by a scalar product.

13 Review of Smooth Manifolds and Vector Bundles 13.1 Smooth Manifolds 13.1.1 Topological Spaces Topology. Let M be a set. A collection S of its subsets is called a topology on M, if it satisfies the following axioms: (S1 ) M and 0 are elements of S. (S2 ) If {O α }α∈I is any family of elements of S, then its union, ⋃α∈I O α , is an element of S. (S3 ) If O1 , . . . , O k are elements of S, then their intersection, ⋂ki=1 O i , is an element of S. Each element O of the set S is called the open subset of M. If F ⊂ M is such a subset of M that M \ F ∈ S, then F is called a closed subset of M. The ordered pair (M, S) is called a topological space [64, 148, 149], and the elements of M are called points. In the entire book, if this does not lead to any misunderstanding, we omit the symbol S and denote the topological space (M, S) simply by M. Metric space. Consider the particular example of a topological space. Let M be a set. A metric on M is a mapping d : M × M → ℝ that satisfies the following axioms: (i) Positivity. ∀p, q ∈ M : d(p, q) ≥ 0. Moreover, d(p, q) = 0 iff p = q. (ii) Symmetry. ∀p, q ∈ M : d(p, q) = d(q, p). (iii) Triangle inequality. ∀p, q, r ∈ M : d(p, r) ≤ d(p, q) + d(q, r). The ordered pair (M, d) is called a metric space [64, 148]. The topology on the metric space can be induced by the metric. Indeed, let p ∈ M be a point and r > 0 be a number. Then the open ball B r (p) of radius r around p is the set B r (p) := {x ∈ M | d(x, p) < r} . Let S d ⊂ 2M be the following collection of subsets O ⊂ M: S d := {O ∈ 2M | ∀p ∈ O ∃r > 0 : B r (p) ⊂ O} . The collection S d satisfies axioms (S1 )–(S3 ). Thus, Sd is a topology on M, called the metric topology. A vector space V with norm ‖ ⋅ ‖ gives the example of metric space, in which the metric is defined by d(u, v) := ‖u − v‖. The space ℝs . The particular case of topological space is one of the main actors in manifold theory; the vector space ℝs , s ∈ ℕ. It is equipped with a norm ‖ ⋅ ‖, such that ‖(x1 , . . . , x s )‖ := √(x1 )2 + ⋅ ⋅ ⋅ + (x s )2 . As for any metric space, the topology on ℝs is https://doi.org/10.1515/9783110563214-013

316 | 13 Review of Smooth Manifolds and Vector Bundles based on the notion of the open ball: if p ∈ ℝs and r > 0, then the open ball B r (p) is defined by B r (p) = {x ∈ ℝs | ‖x − p‖ < r} . Thus, a set O ⊂ ℝs is open iff for any point p ∈ O, there exists such r > 0, that B r (p) ⊂ O. Neighborhoods. If M is a topological space and p ∈ M, then by a neighborhood of p we mean any open set O that contains p [64]. Often, the notion of neighborhood is introduced in a more general sense [148, 149]. That is, V ⊂ M is a neighborhood of p, if p ∈ V and there exists such an open subset O of M that p ∈ O ⊂ M. We do not use such a generalization. Subspace topology. Suppose that (M, S) is a topological space and U ⊂ M is a subset. The topology S defined on M induces the topology SU on U as follows: SU := {V ⊂ U | ∃O ∈ S : V = U ∩ O} . The topology SU is called the subspace topology, and the space (U, SU ) is called the subspace of (M, S). The elementary properties of the subspace follow from its definition: i) A subset V ⊂ U is closed in U if and only if there exists a closed subset F of M, such that V = U ∩ F, ii) If V is open in U and U is open in M, then V is open in M. The statement remains true if “open” is changed everywhere into “closed”. iii) If V ⊂ U is open in M, then it is open in U. The case of closed subset is similar. We use these properties in the definition of the joining operation ∨ (see Chapter 8, Section 8.5.1). Interior. Closure. Boundary. Let (M, S) be a topological space and A ⊂ M. Its interior in M, denoted by Int A, is the maximal open subset of M that contains A: Int A := ⋃{O ∈ S | O ⊂ A} . It follows that Int A ∈ S and Int A ⊂ A. The closure of A in M, denoted by A, is the minimal closed set that contains A: A := ⋂{O ∈ S | A ⊂ M \ O} . The exterior of A, denoted by Ext A, is the interior if its complement: Ext A := Int (M \ A) . The boundary of A, denoted by ∂A, is defined as follows: ∂A := M \ (Int A ∪ Ext A) .

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Note that the interior, exterior, closure, and boundary of a set can be defined in an alternative way. Particularly, the definitions of these notions in terms of the metric are given in [148]. Continuous mappings. The topological structures over two sets allow us to define the notion of continuous mapping between them without referring to the particular properties of ℝ, which may not even be used in the definition of topology on these sets. Such formalism is applicable to situations when topology is not generated by any metric. Let (M, SM ) and (N, SN ) be two topological spaces, and f : M → N be a mapping. Such a mapping is called continuous [64], if for any open set O ⊂ N its pre-image, f −1 (O), is open in M. Denoting the set of all continuous mappings with the domain M and the codomain N by C0 (M; N), one arrives at the following definition: f ∈ C0 (M; N)

⇔

(∀O ∈ SN : f −1 (O) ∈ SM ) .

Inclusion map. Suppose that (M, S) is a topological space and U ⊂ M. The mapping ιU : U → M, which is defined by ∀p ∈ U : ιU (p) := p ,

(13.1)

is called the inclusion map of U in M. The inclusion map is a continuous mapping between topological spaces (U, SU ) and (M, S) [64]. Homeomorphism. If f : M → N is bijective, and both f , f −1 are continuous, then f is called a homeomorphism. Hausdorff space. Conventional analysis of ℝs assumes that any two distinct points possess disjoint neighborhoods. The definition of a topological space given by axioms (S1 )–(S3 ) is so wide that this assumption, in general, is no longer valid. This, in particular, leads to a loss of uniqueness of the convergent sequences’ limits. To avoid such a situation one needs to restrict the set of all possible topologies on M by the following condition: for each pair of distinct points p, q ∈ M, there exist disjoint neighborhoods U of p and V of q, i.e., U ∩ V = 0. The space (M, S), where the topology S satisfies this condition, is called the Hausdorff space [64]. Basis of topology. Through the analogy for the decomposition of an arbitrary vector over some basis, chosen in vector space, one can try to find the subset B of the topology S, which would be much “narrower” than S and permits one to represent any open subset as a union of the elements from B. More precisely, if S is a topology on M, then a collection B of some subsets of M is called a basis for the S, if (B1 ) B ⊂ S; (B2 ) for every open subset O ∈ S there exists such a family {U α }α∈I of elements from B, that ⋃α∈I U α = O. For example, the set B = {B r (x1 , . . . , x s ) | (r ∈ ℚ) ∧ (∀i ∈ {1, . . . , s} : x i ∈ ℚ)} is a basis for the topology on ℝs .

318 | 13 Review of Smooth Manifolds and Vector Bundles Second countable space. A topological space (M, S) is second countable if a countable basis for its topology exists. Let U be a collection of subsets of M. This collection is a cover of M, if ⋃U∈U U = M. The cover is called open, if all its elements are open sets. Any subset U󸀠 ⊂ U, which is a cover of M itself, is called a subcover of U. According to Lindelöf’s theorem, the second countability allows one to extract a countable subcover of any open cover of M. The advantage of such an assertion appears in manifold theory; one can choose a countable set of charts for a complete coordinate description of a manifold. Topological manifolds. The formal definition of topological manifold bears on the idea of covering a set, underlying the manifold, by a system of local coordinates. More strictly, a topological space (M, S) is an s–dimensional topological manifold¹, s ∈ ℕ, [64] if: (M1 ) (M, S) is a Hausdorff space with countable base; (M2 ) for each point p ∈ M one can find such a neighborhood U ∈ S and such an open set O ⊂ ℝs that there exists a homeomorphism φ : U → O between them. The axiom (M2 ) asserts that for M, there exists a set A = {(U α , φ α )}α∈I , such that (A1 ) for each α ∈ I, the ordered pair (U α , φ α ) consists of a non-empty open subset U α of M and a homeomorphism φ α : U α → O α between U α and an open set O α ⊂ ℝs ; (A2 ) ⋃α∈I U α = M. Such a set A is called an atlas on M. The natural number dim M := s defines the dimension of M, each ordered pair (U α , φ α ) specifies a (coordinate) chart on M. We refer to U α as the (coordinate) domain and to φ α as the coordinate map. Since, due to the axiom (M1 ), the space (M, S) is second countable, one can choose a countable index set I. Transition map. The atlas A defines a collection of local coordinates on the manifold. Each point p ∈ M is contained in the domain U α of some chart (U α , φ α ). The coordinate map φ α endows p with an s-tuple (x1 , . . . , x s ) = φ α (p), which represents the local coordinates of the point p. Suppose that for some distinct α, β ∈ I, the coordinate domains U α and U β have a non-empty intersection: U α ∩ U β ≠ 0. The points p from the coordinate domain U α have local coordinates (x1 , . . . , x s ) = φ α (p), while the points p from the coordinate domain U β have local coordinates (̃x1 , . . . , ̃x s ) = φ β (p). In the intersection U α ∩ U β , one has the homeomorphism φ β ∘ φ−1 α | φ α (U α ∩U β ) : φ α (U α ∩ U β ) → φ β (U α ∩ U β ) ,

(13.2)

1 The definition of a topological manifold makes sense even in the case s = 0, but we do not consider it.

13.1 Smooth Manifolds | 319

which is called a transition map. This homeomorphism determines the coordinate transformation (x1 , . . . , x s ) 󳨃→ ̃x i , i = 1, . . . , s, which associates s-tuple (x1 , . . . , x s ) with (̃x1 , . . . , x̃ s ), which gives different coordinate representations for a point p ∈ Uα ∩ Uβ . Open submanifolds. In some cases, one needs to consider mappings which domains are open subsets of manifolds. Mappings that are defined on shapes of the body may serve as an example (we assume here that the dimensions of the body and the physical space coincide). Suppose that (M, S) is a topological s-dimensional manifold with atlas A = {(U α , φ α )}α∈I , and that U ⊂ M is an open set. Consider it with the induced topology. The topological space (U, SU ) is Hausdorff, with a countable base. Let J ⊂ I be a set with the property: ∀α ∈ J : U ∩ U α ≠ 0 . Since each point of U is contained in a coordinate domain of some chart from A, the set J is not empty. Define the set AU = {(U ∩ U α , φ α |U∩U α )}α∈J . For each α ∈ J, the set U ∩ U α is open relative to the topology SU , and the mapping φ α |U∩U α is a homeomorphism onto its image. Thus, AU is an atlas on U, and the topological space (U, SU ) is the topological s-dimensional manifold. We refer to such a space as the open submanifold of M [64]. Product manifolds. In this book, we consider mappings that may be defined on products of topological manifolds. The material velocity gives an example of such a mapping. The topological structure can be defined on the product of topological manifolds as follows. Suppose that (M1 , S1 ) and (M2 , S2 ) are topological manifolds with atlases A1 = {(U α , φ α )}α∈I , A2 = {(V β , ψ β )}β∈J and dimensions s1 and s2 , respectively. Then, the set B = {O1 × O2 | (O1 ∈ S1 ) ∧ (O2 ∈ S2 )} generates the unique topology S on M1 × M2 (called the product topology), for which B is the base [64]. Every set O ∈ S is the union O = ⋃U∈CO U of sets (they are Cartesian products) from some subset CO ⊂ B. It is easy to check that the so-defined topological space (M1 ×M2 , S) is a Hausdorff space with a countable base. One can define an atlas A = {(U α × V β , φ α × ψ β )}α∈I,β∈J on M1 × M2 , where φ α × ψ β : U α × V β → ℝs1 +s2 ,

φ α × ψ β (p, q) := (φ α (p), ψ β (q))

are homeomorphisms onto their images [64]. Thus, (M1 × M2 , S) is a s1 + s2 -dimensional topological manifold. Topological manifolds with boundary. The definition of a topological manifold is useful for the description of a body and a physical space as open sets [37]. However, the formalization of the part of a body requires a more complicated construction. It must contain the information about the interior of the part, treated as a subbody, as well as about the boundary, which is the arena of interaction between the part and the rest

320 | 13 Review of Smooth Manifolds and Vector Bundles

of the body. Such a construction consists of two topological manifolds, where one describes the interior and another describes the boundary. To this end, introduce the set ℍ s , that is, the closed s-dimensional upper half-space: ℍs = {(x1 , . . . , x s ) ∈ ℝs | x s ≥ 0} . Consider ℍ s as the topological space with the topology induced from ℝs . A topological space (M, S) is an s-dimensional topological manifold with boundary, s ∈ ℕ, [64] if: (MB1 ) (M, S) is a Hausdorff space with a countable base; (MB2 ) for each point p ∈ M, one can find such a neighborhood U ∈ S and such a set O, which is either an open subset of ℝs or an open subset of ℍs , that there exists a homeomorphism φ : U → O between them. The axiom (MB2 ), like the corresponding (M2 ), asserts that for M, there exists a set A = {(U α , φ α )}α∈I , where (AB1 ) for each α ∈ I, the ordered pair (U α , φ α ) is such that U α is an open subset of M and φ α : U α → O α is a mapping between U α and a set O α . Such a set is either an open subset of ℝs or an open subset of ℍ s . The mapping φ α is a homeomorphism; (AB2 ) ⋃ α∈I U α = M. The definition for atlas, chart, etc., can be repeated here unchanged. A chart (U α , φ α ) is said to be an interior chart, if φ α (U α ) is an open subset of ℝs , and it is said to be a boundary chart, if φ α (U α ) is an open subset of ℍ s , such that φ α (U α ) ∩ ∂ℍ s ≠ 0. The latter condition specifies that each point from a boundary chart has a local coordinate x s = 0. Such a distinction in charts results in a distinction in points; a point from M is called an interior point of M, if it belongs to the domain of some interior chart, and such a point is called a boundary point of M, if it is an element of the domain of a boundary chart. Such a definition implies the representation of M as a disjoint union M = Int M ⊔ ∂M , in which Int M is the set of all interior points, the interior of M, and ∂M is the set of all boundary points, the boundary² of M [64]. Note that the topological space M, which is endowed with the structure prescribed by axioms (M1 )–(M2 ), is a specific case of a manifold with boundary; all its points are interior ones, that is, ∂M = 0. We will refer to such a manifold as an “sdimensional topological manifold”, while in the common case (∂M ≠ 0), we will use the term “s-dimensional topological manifold with boundary”.

2 Note that such notions are not equivalent to the topological notions of interior and boundary.

13.1 Smooth Manifolds | 321

Suppose that M is an s-dimensional topological manifold with boundary. The interior, Int M, is an open subset of M, and it is an s-dimensional topological manifold. The boundary, ∂M, is a closed subset of M, and it is also a topological manifold, but its dimension is less per unit, i.e., dim ∂M = s − 1 [64].

13.1.2 Smooth Structure Smooth compatibility. The notion of a topological manifold allows one to introduce local coordinates in some neighborhood of any point. Such a notion provides the analytical description for points in a body and a physical space. The structure of the topological manifold, however, is not sufficient for the correct definition of the smoothness of the mappings, specified on its subsets. Therefore, the manifold must be endowed with an additional structure, namely a smooth structure. This allows one to define the notion of a smooth mapping on the manifold independently to a particular choice of coordinates, only relative to the common properties of all atlases in this structure. Suppose that M is a topological manifold with atlas A = {(U α , φ α )}α∈I . Charts (U α , φ α ), (U β , φ β ) on M are called smoothly compatible [3], if one of the following holds: 1) U α ∩ U β = 0; 2) U α ∩ U β ≠ 0, and the transition map (13.2) is a C∞ -diffeomorphism. An atlas A in which any two charts are smoothly compatible with each other is called a smooth atlas. The case of manifolds with boundary is considered in the analogous way. However, some words should be mentioned about transition maps. In the case of manifold s without boundary, the transition map φ β ∘ φ−1 α acts between open subsets of ℝ and its smoothness is understood in the standard sense of calculus (all its component functions have continuous partial derivatives of all orders). In the case of manifold with boundary, the set φ α (U α ∩ U β ) can be open in ℍ s , but not in ℝs . Thus, one requires to generalize the notion of smoothness for φ β ∘ φ−1 α . Suppose that O is an open subset s k of ℍ and f : O → ℝ is some mapping. This mapping is called to be of class C∞ , if for any point x ∈ O there exists an open subset U ⊂ ℝs , that contains x, and a smooth function ̃f : U → ℝk , such that ̃f |U∩ℍs = f|U∩ℍs . In other words, f is smooth, if locally it can be extended to a mapping that is smooth in the ordinary sense. Equivalence relation between smooth atlases. Let M be a topological manifold with or without boundary. Consider the set of all possible atlases on M. Two smooth atlases A = {(U α , φ α )}α∈I , B = {(V β , ψ β )}β∈J on M are equivalent (A ∼ B), if their union, A ∪ B, results in a smooth atlas. This relation satisfies all conditions for equivalent relations for sets. The collection of all smooth atlases can be split into equivalence

322 | 13 Review of Smooth Manifolds and Vector Bundles classes D = [A]. These equivalence classes are called smooth structures on the manifold M [49, 111]. Suppose that D is a smooth structure. The maximal atlas Amax , which corresponds to D, is the union of all smooth atlases that constitute D. Any chart from the maximal atlas is called a smooth chart. The ordered triple (M, S, D) is called a smooth manifold if ∂M = 0 and a smooth manifold with boundary in the opposite case. For the sake of brevity, we suppress the symbols S, D in (M, S, D) and write “M is a smooth manifold” (or, respectively, a “smooth manifold with boundary”).

13.1.3 Smooth Mappings Smooth mappings between smooth manifolds play a primary role in the present book. Configurations, deformations, and energy density distributions are only a few examples of such mappings. Bearing on the notion of smooth structure over a manifold, one can obtain a very common definition for the smooth mappings between them [3, 119]. The real algebra C ∞ (M) of smooth functions. Let (M, S, [A]) be an s-dimensional smooth manifold (with or without boundary). Consider a function f : M → ℝ. Choose some chart (U, φ) ∈ A. The mapping ̃f φ = f ∘ φ−1 : φ(U) → ℝ , acts on the set φ(U) ⊂ ℝs , which is open either in ℝs or in ℍ s . We refer to the map ̃f φ as the coordinate (local) representation of f with respect to the chart (U, φ). Thus, each function f : M → ℝ can be represented locally as a function defined on an open subset of ℝs or ℍ s . The notion of smoothness for such functions was defined in conventional calculus [44]. Taking this into account one can define the notion of a smooth function on a smooth manifold. A function f : M → ℝ is called a smooth function, or of a class C ∞ , if for every p ∈ M, there exists a chart (U, φ) ∈ A such that p ∈ U, and the coordinate representation ̃f φ is smooth on the set φ(U). Denote the set of all smooth functions f : M → ℝ by C∞ (M). Such a set forms a real vector space, if one defines addition and multiplication on scalars pointwise: (f + g) : M → ℝ , (λf) : M → ℝ ,

(f + g)(p) := f(p) + g(p) , (λf)(p) := λf(p) ,

where f, g ∈ C ∞ (M), λ ∈ ℝ. Pointwise multiplication (fg) : M → ℝ ,

(fg)(p) := f(p)g(p) ,

turns C ∞ (M) into commutative and associative algebra³ over ℝ. 3 The definition of algebra is given in Chapter 12, Section 12.5.2.

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| 323

Let (M, S, D) be an s-dimensional smooth manifold. If U ⊂ M is an open subset of M, then, considering it as the open submanifold of M, one can use the notion C ∞ (U) for the set of all smooth functions f : U → ℝ. Choose some smooth chart (U, φ) from the maximal atlas of M. For each i = 1, . . . , s define the projection map π is : ℝs → ℝ ,

π is (x1 , . . . , x s ) := x i .

Choose one of i ∈ {1, . . . , s} and construct the scalar-valued function φ i := π is ∘ φ : U → ℝ . The coordinate representation of this function is ̃ φi

=

(13.3) φ i ∘ φ−1 :

φ(U) → ℝ. Obviously,

̃ φ i = π is |φ(U). Since π is is smooth, ∀i = 1, . . . , s : φ i ∈ C∞ (U) . Smooth mappings between manifolds. Suppose that (M, SM , [A]) and (N, SN , [B]) are smooth manifolds (with or without boundary) with dimensions dim M = s and dim N = k. Consider a continuous mapping 𝜘 : M → N. Choose some chart (V, ψ) ∈ B. One can find at least one chart (U, φ) ∈ A, such that U ∩ 𝜘−1 (V) ≠ 0. Due to the continuity of 𝜘 the set U ∩ 𝜘−1 (V) is open in U and, hence, the set φ(U ∩ 𝜘−1 (V)) is open either in ℝs or in ℍs . The mapping ̃ φ,ψ = ψ ∘ 𝜘 ∘ φ−1 : φ(U ∩ 𝜘−1 (V)) → ψ(V) 𝜘 is called the coordinate (local) representation of 𝜘 with respect to the charts (U, φ) and (V, ψ). It is a map between subsets of ℝs and ℝk . Let (x j ) denote local coordinates ̃ φ,ψ can be represented for M and (y i ) denote local coordinates for N. Then, the map 𝜘 by k mappings y i : (x1 , . . . , x s ) 󳨃→ y i (x1 , . . . , x s ) ,

i = 1, . . . , k .

A mapping 𝜘 : M → N is called a smooth mapping or of a class C ∞ if it is continuous and for every p ∈ M, there exist charts (U, φ) ∈ A and (V, ψ) ∈ B, such that p ∈ U, ̃ φ,ψ is of class C∞ . In this book, we U ∩ 𝜘−1 (V) ≠ 0 and the coordinate representation 𝜘 denote the set of all smooth mappings 𝜘 : M → N by C∞ (M; N). Diffeomorphism. Let (M, SM , [A]), and (N, SN , [B]) be smooth manifolds (with or without boundary). A mapping 𝜘 : M → N is called a C∞ -diffeomorphism (or just a diffeomorphism) if it is bijective, and both 𝜘, 𝜘−1 are of class C∞ . Consider an example. Suppose that (M, S, [A]) is a smooth s-dimensional manifold. Choose some chart (U, φ) ∈ A. The mappings φ i (13.3), i = 1, . . . s, are smooth. Now consider the mapping φ : U → O. Here, O = φ(U) is an open submanifold of ℝs with the chart (O, Id O ). The coordinate representation of φ has the form ̃ φ φ,IdO = IdO ∘ φ ∘ φ−1 = IdO . This mapping is of class C∞ , and we have that φ ∈ C∞ (U; O). Since the mapping φ is a homeomorphism, the inverse map φ−1 : O → U exists. Its coordinate representation is similar to IdO . Thus, φ : U → O is a diffeomorphism between smooth manifolds that are represented by U and O.

324 | 13 Review of Smooth Manifolds and Vector Bundles Immersions and embeddings. Let (M, SM , [A]) and (N, SN , [B]) be smooth manifolds with dimensions dim M = s and dim N = k. Among smooth mappings from C∞ (M; N), there are mappings with specific properties that play an important role in the definitions of configurations and deformations. We will analyze them in detail. Suppose that 𝜘 ∈ C∞ (M; N) and p ∈ M. Choose some charts (U, φ) ∈ A, (V, ψ) ∈ B, such that p ∈ U, U ∩ 𝜘−1 (V) ≠ 0, and consider the map ̃ φ,ψ : (x1 , . . . , x s ) 󳨃→ (y1 , . . . , y k ) . 𝜘 One can calculate the Jacobian matrix of this map at point φ(p), that is, the matrix with elements 󵄨 ∂y i 󵄨󵄨󵄨 󵄨󵄨 , i = 1, . . . , k , j = 1, . . . , s . ∂x j 󵄨󵄨󵄨φ(p) Next, one can calculate the number of linearly independent rows (or columns) in this matrix. This number is called the rank of 𝜘 at p and is denoted by rank|p 𝜘. We restrict ourselves to the case s ≤ k. Then, a map 𝜘 ∈ C∞ (M; N) is called a smooth immersion if its rank is constant and is equal to rank|p 𝜘 = s for every point p ∈ M. If, in addition, 𝜘 is a homeomorphism onto its image, 𝜘(M), then 𝜘 is called a smooth embedding. Partitions of unity. For a global definition of the function on a manifold that is covered by several maps, one has to apply some technique for gluing coordinate representations in different charts. This technique can be developed bearing on the topological concept of the partition of unity. In particular, it allows one to define the integration over an arbitrary subset of a manifold. Suppose that M is a smooth manifold with or without boundary. The support supp f of a mapping f : M → ℝ is the closure of the set f −1 (ℝ \ {0}): supp f = f −1 (ℝ \ {0}) . A partition of unity on a smooth manifold M [123] is a family {(U α , f α )}α∈I that satisfies the following conditions: i) The family {U α }α∈I is a locally finite open cover of⁴ M. ii) For each α ∈ I, one has that f α ∈ C∞ (M) and supp f α ⊂ U α . iii) For all α ∈ I and for each x ∈ M: f α (x) ≥ 0. iv) for all x ∈ M, one has ∑ f α (x) = 1. α∈I

By i) the sum in iv) is finite. If A = {(V α , φ α )}α∈I is an atlas on M, then a partition of unity, subordinated to A, is such a partition of unity {(U β , f β )}β∈J , that for any β ∈ J there exists α ∈ I with the property U β ⊂ V α .

4 That is, any point from M has a neighborhood that intersects only with a finite number of U α .

13.1 Smooth Manifolds

| 325

13.1.4 Embedded Submanifolds The embedding of a smooth manifold into another smooth manifold, probably, with higher dimension, defines a subset of it. This subset can be endowed with the structure of a smooth manifold also. If this structure is aligned with the structure of an enveloping manifold, then the image of the embedded manifold becomes the submanifold of the enveloping one. Such a way to define submanifolds is intensively used in continuum mechanics. For example, the shape of a body is, on the one hand, an image of a material manifold and, on the other hand, a submanifold of the physical space. Another example is given by the definition of the body part, which is a smooth manifold with boundary, aligned with a smooth structure of the whole body (material manifold). Below, we intend to briefly treat submanifolds by following [3]. Definition of a submanifold. Let M be a smooth manifold with or without boundary. A smooth manifold (with or without boundary) N ⊂ M is called an embedded submanifold of M if (i) N is endowed with the subspace topology; (ii) the smooth structure of N is such that the inclusion map ιN : N → M (13.1) is a smooth embedding. An open submanifold is an example of an embedded submanifold. Conversely, if M is a smooth manifold, and N ⊂ M is an embedded submanifold of M such that dim N = dim M, then N is an open subset of M. An image as a submanifold. Consider two smooth manifolds M, N with dimensions dim M ≤ dim N. Let 𝜘 : M → N be a smooth embedding. The set 𝜘(M) can be endowed with topological and smooth structures as follows. The topology S𝜘(M) on 𝜘(M) is induced from N. Let A = {(U α , φ α )}α∈I be an atlas from the smooth structure of M. Then, the family {𝜘(U α )}α∈I is an open cover of 𝜘(M) (in the induced topology of 𝜘(M)). Since 𝜘 is a smooth embedding, the mapping ̂ : M → 𝜘(M) , 𝜘

̂ (p) := 𝜘(p) , 𝜘

is a homeomorphism. For α ∈ I one has that φ α : U α → O α is a homeomorphism and ̂ −1 : 𝜘(U α ) → O α is a homeomorphism. Suppose that α, β ∈ I in this regard, ψ α = φ α ∘ 𝜘 are distinct indices with 𝜘(U α ) ∩ 𝜘(U β ) ≠ 0. Then, U α ∩ U β ≠ 0 and φ β ∘ φ−1 α | φ α (U α ∩U β ) : φ α (U α ∩ U β ) → φ β (U α ∩ U β ) , is a C∞ -diffeomorphism. Hence, ̂ −1 ∘ 𝜘 ̂ ∘ φ−1 ψ β ∘ ψ−1 α = φβ ∘ 𝜘 α , is also a C∞ -diffeomorphism. It proves that A𝜘(M) = {(𝜘(U α ), ψ α )}α∈I is a smooth atlas on 𝜘(M). Therefore, (𝜘(M), S𝜘(M) , [A𝜘(M) ]) is a smooth manifold.

326 | 13 Review of Smooth Manifolds and Vector Bundles ̂ : M → 𝜘(M). In the appropriate charts (U α , φ α ) and Consider the mapping 𝜘 (𝜘(U α ), ψ α ), one has ̃̂ = ψ α ∘ 𝜘 ̂ ∘ φ−1 𝜘 α = Id O α . ̂ −1 , one obtains Similarly, for 𝜘 ̃ ̂ −1 ∘ ψ−1 ̂ −1 = φ α ∘ 𝜘 𝜘 α = Id O α . ̂ is a C∞ -diffeomorphism. Moreover, the inclusion map ι𝜘(M) : 𝜘(M) → N is Hence, 𝜘 a composition ̂ −1 , ι𝜘(M) = 𝜘 ∘ 𝜘 of a diffeomorphism and a smooth embedding and, consequently, ι𝜘(M) is a C∞ embedding. As a result, we conclude that triple (𝜘(M), S𝜘(M) , [A𝜘(M) ]) defines an embedded submanifold of N. Restricting maps to submanifolds. Suppose that M, N are smooth manifolds, and 𝜘 : M → N is a smooth mapping. If U ⊂ M is an embedded submanifold, then 𝜘|U = 𝜘∘ ιU : U → N is a composition of smooth mappings. Therefore, the restriction 𝜘|U : U → N is smooth.

13.2 The “Tower” of Tensor Spaces One can use Whitney’s embedding theorem and consider the embedding of a smooth manifold (a body or a physical space) in some ambient Euclidean space of higher dimension. The representation of a smooth manifold as a hypersurface in a Euclidean space allows one to introduce tangent vectors on it as directed line segments by appealing to the generalization of conventional Euclidean geometry on higher dimensions. However, such a direct approach turns out to be cumbersome because of the higher dimension of the ambient space. At the same time, the description of geometric objects by methods of internal geometry has always determined the elegance of the geometric theory. In this regard, it seems more convenient to define the tangent vectors, covectors, etc., as intrinsic objects on smooth manifolds.

13.2.1 Tangent Space to a Smooth Manifold Suppose that M is a smooth s-dimensional manifold and p ∈ M is a point. Intuitively, a tangent vector to M at p represents an “infinitesimal”, or, in other words, “local”, object. Such an object must be introduced in an intrinsic manner, that is, in terms of the “inhabitants” of a smooth manifold; the smooth curves and smooth functions defined on it. One can formalize the notion of locality by dealing with smooth curves and smooth functions as follows.

13.2 The “Tower” of Tensor Spaces | 327

Tangent vectors as an equivalence class of curves. A smooth curve on M is a mapping χ ∈ C∞ (𝕁; M), where 𝕁 ⊂ ℝ is a real line interval (open or closed). A curve χ ∈ C∞ (𝕁; M) passes through the point p ∈ M, if there exists c ∈ 𝕁, such that χ(c) = p. Consider the set Cup (M) of all smooth curves that satisfy the following conditions: χ ∈ Cup (M) iff (Cu1 ) χ is a smooth curve on M that is defined on an open interval containing 0. (Cu2 ) χ(0) = p. Moreover, χ −1 ({p}) = {0}. The condition χ −1 ({p}) = {0} requires that 0 ∈ ℝ is the unique point that transforms to p. Let (U, φ) be a chart from the maximal atlas of M that contains p. Introduce the following equivalence relation ∼p [119]: ∀χ 1 , χ2 ∈ Cu p (M) : (χ 1 ∼p χ 2 ) ⇔ (

d(φ ∘ χ 1 )(t) 󵄨󵄨󵄨󵄨 d(φ ∘ χ 2 )(t) 󵄨󵄨󵄨󵄨 󵄨󵄨 = 󵄨󵄨 ) . 󵄨󵄨t=0 󵄨󵄨t=0 dt dt

Note that φ ∘ χ i , i = 1, 2, are smooth mappings from symmetric intervals to ℝs , and the derivative d/dt is understood in the sense of conventional calculus [44]. If (V, ψ) is another chart that contains p, then by the chain rule one could be assured that if χ 1 ∼p χ 2 relatively to the chart (U, φ), then χ1 ∼p χ 2 relatively to (V, ψ). Thus, the equivalence relation does not depend on a particular chart. Each equivalence class u = [χ]p is called a tangent vector at p, and the quotient set T p M := Cup (M)/ ∼p is called the tangent space to M at p [119]. The vector structure of the tangent space. Every chart (U, φ) from the maximal atlas of M, which contains p, generates the mapping φ

c p : T p M → ℝs ,

φ

cp : u = [χ]p 󳨃→

d(φ ∘ χ) 󵄨󵄨󵄨󵄨 󵄨 = (u 1 , . . . , u s ) . dt 󵄨󵄨󵄨t=0

(13.4)

By the definition of the equivalence relation ∼p , such a mapping is well defined, i.e., φ its value does not depend on a representative χ ∈ u. Moreover, the mapping cp is a bijection. This can be easily verified by means of the following arguments. φ φ Suppose that u 1 , u 2 ∈ T p M and u 1 ≠ u 2 . If one assumes that cp (u 1 ) = cp (u 2 ) then, choosing representatives χ 1 ∈ u 1 and χ 2 ∈ u 2 , one obtains that χ 1 ∼p χ 2 . This φ implies a contradiction with u 1 ≠ u 2 . Thus, cp is injective. 1 s s Now, let (u , . . . , u ) ∈ ℝ be an arbitrary s-tuple and (p1 , . . . , p s ) = φ(p). Choose a > 0, such that the image ̃χ(] − a, a[) of the curve, ̃χ ∈ C∞ (] − a, a[; ℝs ) ,

̃χ(t) := (p1 + u 1 t, . . . , p s + u s t) ,

is contained in φ(U). Taking χ = φ−1 ∘ ̃χ , one obtains the curve χ ∈ Cu p (M), such that φ φ cp ([χ]p ) = (u 1 , . . . , u s ). Thus, cp is surjective. We have arrived at the desired concluφ sion: cp is bijective.

328 | 13 Review of Smooth Manifolds and Vector Bundles φ

By definition, the mapping cp depends on the chart (U, φ). Let (U, φ) and (V, ψ) be two charts from the maximal atlas that contain p. The chain rule implies that cp = D φ(p) (ψ ∘ φ−1 ) ∘ cp , ψ

φ

(13.5)

where D φ(p) (ψ ∘ φ−1 ) ∈ Lin(ℝs ; ℝs ) is the total derivative of the transition map ψ ∘ φ−1 at φ(p). φ Since cp : T p M → ℝs is a bijective mapping with values on the s-dimensional vector space ℝs , one can induce the real vector space structure of the latter into T p M φ by means of cp . To this end, define the mappings φ −1

+ : Tp M × Tp M → Tp M ,

(u + v) := (cp )

⋅ : ℝ × Tp M → Tp M ,

(λ ⋅ u) := (cp )

φ −1

φ

φ

(cp (u) + cp (v)) , φ

(λcp (u)) ,

φ

where u, v ∈ T p M and λ ∈ ℝ. Although the mapping cp depends on the chart (U, φ), direct calculations taking the equality (13.5) into account show that the introduced mappings (+) and (⋅) are chart-independent. Thus, the mappings (+) and (⋅) are a wellφ defined real vector space structure on T p M, and cp is the vector space isomorphism relative to such a structure. This implies that dim T p M = s. Coordinate basis. The standard basis of ℝs is formed by the collection of the s-tuples (δ1i , . . . , δ si ), where 1 stands in the i-th position and 0 stands in others. Choose some φ smooth chart (U, φ), which contains p. Since c p is the vector space isomorphism, the collection (e i , . . . , e s ) of tangent vectors φ −1

e i = (cp )

(δ1i , . . . , δ si ) ∈ T p M ,

i = 1, . . . , s ,

(13.6)

forms a basis in T p M. Each vector e i is an equivalence class of the curve χ i , such that φ ∘ χ i (t) = (p1 + δ1i t, . . . , p s + δ si t) ,

(13.7)

where (p1 , . . . , p s ) = φ(p). We refer to the basis (e i , . . . , e s ) as coordinate basis. Such a basis depends on the chart (U, φ). Each tangent vector u ∈ T p M has a decomposition φ u = u i e i , where (u 1 , . . . , u s ) = cp (u). Suppose that (U, φ) and (V, ψ) are two smooth charts that contain p and establish ̃ i ) are components of the same vector u ∈ the local coordinates (x i ), (̃x i ). If (u i ) and (u T p M in these charts, then the equality (13.5) implies that ũ i =

󵄨 ∂ ̃x i 󵄨󵄨󵄨 󵄨󵄨 uj , ∂x j 󵄨󵄨󵄨φ(p)

i = 1, . . . , s .

(13.8)

Here, ψ ∘ φ−1 : (x1 , . . . , x s ) 󳨃→ (̃x1 , . . . , ̃x s ) is a transition map (13.2). The tangent vector as a derivation of the space of germs. Let U ⊂ M be an open set and f ∈ C∞ (U; ℝ) be a smooth function. The ordered pair (f, U) is called a smooth function element on M [3].

13.2 The “Tower” of Tensor Spaces |

329

Denote the set of all smooth function elements (f, U) on M, where p ∈ U, by Felp (M). Define the equivalence relation ∼p on Felp (M) by the following condition: (f, U) ∼p (g, V), if there exists an open set W ⊂ U ∩ V containing p such that f|W = g|W . The equivalence class f = [(f, U)]p of a function element (f, U) is called the germ of f at p [3], and the quotient set C∞ (M) := Felp (M)/ ∼p represents the set of all germs of smooth functions at p. Note that if U and V are neighborhoods of p, V ⊂ U and (f, U) ∈ Felp (M), then (f|V , V) ∈ [(f, U)]p . In this regard, so that there will not be any confusion, we will write [f]p instead of [(f, U)]p . The set C∞ (M) is endowed by the real vector space structure as follows. Define the binary operations + : C∞ (M) × C∞ (M) → C∞ (M) , ∞

f + g := [(f + g, U ∩ V)]p ,

∞

⋅ : ℝ × C (M) → C (M) ,

λf := [(λf + g, U)]p ,

where (f, U) ∈ f, (g, V) ∈ g and λ ∈ ℝ. It is easy to check that these operations are well defined and bring C ∞ (M) to the real vector space. Moreover, the operation ⋅ : C∞ (M) × C∞ (M) → C∞ (M) ,

f + g := [(fg, U ∩ V)]p ,

C∞ (M)

turns into real associative algebra. Since for arbitrary (f, U), (g, V) ∈ f, the equality f(p) = g(p) holds, the notion f(p) := f(p) is well defined. A derivation of C ∞ (M) [3, 48, 132] is a linear map v : C∞ (M) → ℝ that satisfies the following product rule: ∀f, g ∈ C∞ (M) : v(fg) = f(p)v(g) + g(p)v(f) . Denote the set of all derivations by D p M. Each element of D p M is called a tangent vector at p [3, 48, 132]. This gives an alternative definition of a tangent vector. Suppose that u, v ∈ D p M and λ ∈ ℝ. Since u and v are linear mappings, the sum (u + v) : C∞ (M) → ℝ and the product (λu) : C∞ (M) → ℝ are also linear mappings. Direct calculations show that they satisfy the product rule. Hence, the set D p M is a real vector space. Let v ∈ D p M. The following elementary properties follow directly from the definition of the derivation: ∙ If f is a germ of a constant function then⁵ v(f) = 0. ∙ If f(p) = g(p) = 0 then v(fg) = 0. 5 If f : U → ℝ is such that f(U) = {c}, then [f]p = c[e]p , where e ∈ C ∞ (M) is such function that e(M) = {1}. Since v([f]p ) = cv([e]p ), it is sufficient to prove that v([e]p ) = 0. Using the equality [e]p [e]p = [e]p and the product rule one obtains that v([e]p ) = v([e]p [e]p ) = 2v([e]p ). This implies the desired equality.

330 | 13 Review of Smooth Manifolds and Vector Bundles The isomorphism T p M ≅ D p M. Both equivalence classes of curves and derivations of germs are no more than alternative formalizations for the idea of locality. Despite the fact that such formalizations are constructed with different argumentations, they represent a unified object (up to natural isomorphism). Let u ∈ T p M be a tangent vector. It generates the mapping Lu : C∞ (M) → ℝ, Lu (f) :=

d(f ∘ χ) 󵄨󵄨󵄨󵄨 , 󵄨 dt 󵄨󵄨󵄨t=0

(13.9)

where χ ∈ u and (f, U) ∈ f. The chain rule and the definitions of f and u imply that the value of Lu does not depend on representatives from u and f. It follows from the definition that (13.9) does not require any chart. Thus, Lu is well defined. Since d/dt is linear and satisfies the Leibniz rule, Lu ∈ D p M. Consider the mapping L : T p M → D p M, L : u 󳨃→ Lu , where Lu is defined by (13.9). Such a mapping is the desired isomorphism, because φ ∙ The mapping L is linear. It follows from the linearity of cp . ∙ The mapping L is injective. Suppose that Lu1 = Lu2 . Choose a smooth chart (U, φ) that contains p. Take φ i := π is ∘ φ (13.3), where π is : ℝs → ℝ ,

π is (x1 , . . . , x s ) = x i ,

i = 1, . . . , s .

Then one has Lu1 ([φ i ]p ) = Lu2 ([φ i ]p ), for all i = 1, . . . , s. Choosing representatives χ1 ∈ u 1 , χ 2 ∈ u 2 , one obtains 󵄨 󵄨 d (π is ∘ φ ∘ χ2 ) 󵄨󵄨󵄨 d (π is ∘ φ ∘ χ1 ) 󵄨󵄨󵄨 󵄨󵄨 = 󵄨󵄨 , 󵄨󵄨 󵄨󵄨 dt dt 󵄨󵄨 󵄨󵄨 󵄨t=0 󵄨t=0 ∙

i = 1, . . . , s ,

and this implies u 1 = u 2 . The mapping L is surjective. Suppose that v ∈ D p M. One needs to obtain such tangent vector u ∈ T p M, for which Lu = v. Choose a smooth chart (U, φ), that contains p and denote φ i = π is ∘ φ and (p1 , . . . , p s ) = φ(p). ̃ p ∈ C∞ (M). Without any loss of generality, one can asTake a germ f = [(f, U)] ̃ ⊂ U. Using Taylor’s formula, one obtains the following decomposisume that U ̃ tion on φ(U): f ∘ φ−1 (x1 , . . . , x s ) = f(p) +

󵄨 ∂(f ∘ φ−1 ) 󵄨󵄨󵄨 󵄨󵄨 (x i − p i ) 󵄨󵄨 ∂x i 󵄨φ(p)

+ f ij (x1 , . . . , x s )(x i − p i )(x j − p j ) ,

(13.10)

̃ Identifying f(p) and p i where f ij is a smooth function (the remainder term) on φ(U). ̃ ̃ with the corresponding constant mappings on U, one can rewrite (13.10) on U: f = f(p) +

󵄨 ∂(f ∘ φ−1 ) 󵄨󵄨󵄨 󵄨󵄨 (φ i − p i ) + ̃f ij (φ i − p i )(φ j − p j ) , 󵄨󵄨 ∂x i 󵄨φ(p)

(13.11)

13.2 The “Tower” of Tensor Spaces |

331

where ̃f ij = f ij ∘ φ. Using the algebraic structure of germs, the properties of v, and the relation (13.11), one obtains v([f]p ) =

󵄨 ∂(f ∘ φ−1 ) 󵄨󵄨󵄨 󵄨󵄨 v ([φ i ]p ) . 󵄨󵄨 ∂x i 󵄨φ(p)

(13.12)

The formula (13.12) was derived for an arbitrary germ [f]p . Define a smooth curve χ : t 󳨃→ χ(t) by the relation χ(t) = φ−1 (p1 + tv ([φ1 ]p ) , . . . , p s + tv ([φ s ]p )) . Here, (U, φ) and p i are the same as above. Such a curve belongs to Cu p (M). Set u := [χ]p ∈ T p M. Choose a germ [f]p ∈ C∞ (M). Direct calculations show that Lu ([f]p ) =

󵄨 d(f ∘ χ) 󵄨󵄨󵄨󵄨 d(φ ∘ χ) 󵄨󵄨󵄨󵄨 ∂(f ∘ φ−1 ) 󵄨󵄨󵄨 󵄨󵄨 v ([φ i ]p ) . 󵄨󵄨 = D φ(p) (f ∘ φ−1 ) 󵄨󵄨 = 󵄨󵄨 dt 󵄨󵄨t=0 dt 󵄨󵄨t=0 ∂x i 󵄨φ(p)

One obtains that v(f) = Lu (f) for each germ f ∈ C∞ (M). Thus, u is the desired vector. The above reasonings show that L is a natural isomorphism. Finally, we can claim that T p M ≅ D p M. Suppose that (U, φ) is a smooth chart that contains p, and (e 1 , . . . , e s ) is the corresponding coordinate basis (13.6) of T p M, which is generated by curves (13.7). Since L is the isomorphism, then (Le1 , . . . , Le s ) is the basis in D p M. Thus, if u = u i e i , then Lu = u i Le i . Let [f]p ∈ C∞ (M). Then, Le i ([f]p ) =

󵄨 d(f ∘ χ) 󵄨󵄨󵄨󵄨 ∂(f ∘ φ−1 ) 󵄨󵄨󵄨 󵄨󵄨 . 󵄨󵄨 = D φ(p) (f ∘ φ−1 ) (δ1i , . . . , δ si ) = 󵄨󵄨 dt 󵄨󵄨t=0 ∂x i 󵄨φ(p)

The derivation Le i acts like a partial derivative. This motivates the following notation: ∂ i |p := Le i ,

i = 1, . . . , s .

(13.13)

Then, for a tangent vector u ∈ T p M, one has Lu = u i ∂ i |p . In the following, we identify u ∈ T p M with its image Lu ∈ D p M by virtue of the isomorphism L. Thus, (e i )si=1 is identified with (∂ i |p )si=1 and one obtains the following expansion: u = u i ∂ i |p for a tangent vector u ∈ T p M. Suppose that (U, φ) and (V, ψ) are two smooth charts that contain p and establish local coordinates (x i ), (̃x i ). The relations (13.8) imply that ∂ x i |p =

󵄨 ∂ ̃x j 󵄨󵄨󵄨 󵄨󵄨 ∂ x̃ j |p , ∂x i 󵄨󵄨󵄨φ(p)

i = 1, . . . , s ,

(13.14)

332 | 13 Review of Smooth Manifolds and Vector Bundles where (x1 , . . . , x s ) 󳨃→ (x̃1 , . . . , x̃ s ) is the transition map ψ ∘ φ−1 . Symbols x and x̃ in ∂ x i |p and ∂ x̃ i |p indicate that we are dealing with different coordinate bases. In this book, we use such notations if there is any danger of confusion. An equivalence class of a curve or, similarly, a derivation of germs, represents an “infinitesimal” object at p. The set of all such objects, the tangent space T p M, is the formalization of the idea of an “infinitesimal” neighborhood of p. The tangent vector as a derivation of the algebra C ∞ (M). A derivation at p is a linear map v : C∞ (M) → ℝ that satisfies the “product rule” relation [3, 119]: ∀f, g ∈ C∞ (M) : v(fg) = f(p)v(g) + g(p)v(f) . The set of all derivations at point p can be endowed with the structure of a vector space by means of the operations of addition and scalar multiplication, the same as defined previously for linear mappings. Denote this space by V p M. For any derivation v ∈ V p M, the following elementary properties hold: ∙ If f ∈ C∞ (M) is a constant function then v(f) = 0. ∙ If f(p) = g(p) = 0 then v(fg) = 0. Suppose that f, g ∈ C ∞ (M) are such mappings that f|W = g|W for some neighborhood W ⊂ M of p. Then, for any v ∈ C∞ (M), we have v(f) = v(g) just by the definition of a germ at p. A similar locality property still holds for any v ∈ V p M: v(f) = v(g), but contrary to the case of germ, this property needs to be proved. The proof of such a property based on the construction of a function h : M → ℝ such that h(p) = 1 and h|M\W = 0 is given in [119]. Any vector u ∈ T p M defines the derivation Lu : C∞ (M) → ℝ in much the same way as was the case for germs (see (13.9)): d(f ∘ χ) 󵄨󵄨󵄨󵄨 , Lu (f) := 󵄨 dt 󵄨󵄨󵄨t=0 where χ ∈ u. Reasonings similar to those for constructing the natural isomorphism T p M ≅ D p M allow one to construct the natural isomorphism T p M ≅ V p M.

13.2.2 The tangent Map at a Point Suppose that M and N are s and k-dimensional smooth manifolds, respectively. Let p ∈ M be a point and 𝜘 : M → N be a smooth mapping. This mapping transforms all points of M into points of N. The idea of the tangent map is to define the linear transformation of T p M to T𝜘(p) N, which is aligned with given 𝜘. Below, two alternative definitions are given [3, 123]. ∙ The first definition. Tangent map to 𝜘 at point p is the mapping T p 𝜘 : T p M → T𝜘(p) N , ∀u ∈ T p M : T p 𝜘(u) := [𝜘 ∘ χ]𝜘(p) ,

13.2 The “Tower” of Tensor Spaces | 333

∙

where χ ∈ u. The chain rule implies that such a definition does not depend on the representative of u. Thus, T p 𝜘 transforms the equivalence class of a curve into the equivalence class of the image of this curve. The second definition. The tangent map to 𝜘 at point p is the mapping T p 𝜘 : D p M → D𝜘(p) N , ∀v ∈ T p M : T p 𝜘(v)(f) := v([f ∘ 𝜘]p ) , where f ∈ f ∈ C∞ (N).

The so-defined tangent map satisfies the following properties [3]. Let M, N, P be smooth manifolds, 𝜘 ∈ C∞ (M; N), γ ∈ C∞ (N; P) are smooth mappings, and p ∈ M. Then: i) The mapping T p 𝜘 : T p M → T𝜘(p) N is linear. ii) The tangent map to a composition is a composition of tangent maps: T p (γ ∘ 𝜘) = T𝜘(p) γ ∘ T p 𝜘 : T p M → T γ∘𝜘(p) P . iii) T p IdM = IdT p M . iv) If 𝜘 is a diffeomorphism then T p 𝜘 : T p M → T𝜘(p) N is isomorphism and (T p 𝜘)−1 = T𝜘(p) 𝜘−1 . Such properties are similar to the corresponding properties of the total derivative D x f of a smooth mapping f : ℝs → ℝk at point x. The property i) follows from the chain φ rule applied to total derivative and from the definition of cp . The properties ii) and iii) follow from the definition of the tangent map. Finally, the property iv) follows from ii) and iii).

13.2.3 Cotangent Space to a Smooth Manifold Let M be a smooth s-dimensional manifold and p ∈ M be a point. A real vector space of linear functionals (covectors) over T p M (the dual vector space) is denoted by T ∗p M := (T p M)∗ and is called the cotangent space to M at p. Like in Chapter 12, Section 12.5.2, we define the bilinear operation ⟨⋅, ⋅⟩p : T ∗p M × T p M → ℝ ,

⟨ν, u⟩p := ν(u) .

It follows from the natural isomorphism between T p M and (T p M)∗∗ that ⟨⋅, ⋅⟩p is symmetric: ∀ν ∈ T ∗p M ∀u ∈ T p M : ⟨ν, u⟩p = ⟨u, ν⟩p . The coordinate dual basis. Choose a smooth chart (U, φ), such that p ∈ U. To this φ chart there corresponds the coordinate basis (∂ i |p )si=1 . The mapping cp (13.4) generates

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s linear functionals φ

dx i |p = π is ∘ cp : T p M → ℝ ,

i = 1, . . . , s ,

(13.15)

that satisfy the following conditions: if u = u i ∂ i |p ∈ T p M, then dx i |p (u) = u i . Thus, ⟨dx i |p , ∂ j |p ⟩p = δ ij ,

i, j = 1, . . . , s ,

and (dx i |p )si=1 is a dual basis. We refer to it as the coordinate dual basis. Each covector ν ∈ T ∗p M can be uniquely decomposed as ν = ν i dx i |p . Suppose that (U, φ) and (V, ψ) are two smooth charts with local coordinates (x i ), (̃x i ) that contain p. Therefore, an arbitrary covector ν ∈ T ∗p M can be decomposed in two ways: ν = ν i dx i |p = ̃ν i d ̃x i |p . The formula (13.14) implies that 󵄨 ∂ x̃ j 󵄨󵄨󵄨 󵄨󵄨 ̃ν j , i = 1, . . . , s . νi = (13.16) ∂x i 󵄨󵄨󵄨φ(p) The differential. The differential of a smooth function provides an example of a covector. Suppose that f ∈C∞ (M). The differential of f at p [3] is a mapping df p : T p M→ℝ that acts as follows: ∀u ∈ T p M : df p (u) := u([f]p ) , (13.17) where [f]p is the germ of f at p. It is a linear mapping and, thus, the element of T ∗p M. Using the dual basis (13.15) one obtains 󵄨 ∂(f ∘ φ−1 ) 󵄨󵄨󵄨 󵄨󵄨 dx i |p . df p = 󵄨󵄨 ∂x i 󵄨φ(p) Let (U, φ) be a smooth chart. For i = 1, . . . , s, consider the differential dφ ip of the mapping φ i (13.3) at some point p ∈ U. Since φ i ∘ φ−1 = π is , one has 󵄨 ∂π is 󵄨󵄨󵄨 󵄨󵄨 dφ ip = dx j |p = δ ij dx j |p . ∂x j 󵄨󵄨󵄨φ(p) Hence, dφ ip = dx i |p . This formula means that the covector dx i |p can be considered as the result of applying of the operation d to the map φ i . For a point x, the latter returns its i-th coordinate x i . Such an observation motivates the use of the notation dx i |p for covectors which constitute the coordinate dual basis.

13.2.4 Remarks on the Tangent Spaces The tangent space to an open submanifold. Suppose that M is a smooth manifold and U ⊂ M is an open subset. Let p ∈ U be a point. One has two tangent spaces: T p M and T p U, where U is considered as an open submanifold of M. The tangent map, T p ιU : T p U → T p M to the inclusion map ιU : U → M (13.1) is a vector space isomorphism [3]. Since this isomorphism is basis and chart independent, it is natural, and one can identify any vector u ∈ T p U with its image, T p ιU (u), in T p M. In this entire book we make such an identification.

13.2 The “Tower” of Tensor Spaces | 335

The tangent space to the product manifold. Suppose that M1 and M2 are smooth manifolds. Then, their direct product, M1 × M2 , is a smooth manifold, similarly. Let (p1 , p2 ) ∈ M1 × M2 be a point and T(p1 ,p2 ) M1 × M2 be the tangent space at (p1 , p2 ), defined in the usual manner. Let pri : M1 × M2 → Mi ,

pri (q1 , q2 ) := q i ,

i = 1, 2 ,

be projections. Then the map π : T(p1 ,p2) M1 × M2 → T p1 M1 ⊕ T p2 M2 , ∀u ∈ T(p1 ,p2 ) M1 × M2 : π(u) := (T(p1 ,p2 ) pr1 (u), T(p1 ,p2 ) pr2 (u)) , is a natural isomorphism [3]. This observation allows us to consider the tangent vectors to a product manifold simply as ordered pairs of tangent vectors to each multiplier. Velocity vectors of curves. Any tangent vector represents an equivalence class of smooth curves. In this regard, it can be interpreted as a generalized velocity vector along a curve. Let M be a smooth manifold. Suppose that χ : 𝕁 → M is a smooth curve that is defined on an open interval 𝕁 ⊂ ℝ. Let t0 ∈ 𝕁 be a point. The space 𝕁 is a one-dimensional smooth manifold, and, thus, the coordinate basis of T t 0 𝕁 consists of one vector ∂ t |t 0 . The velocity of χ at t0 is the vector χ󸀠 (t0 ) ∈ T χ(t 0) M, χ 󸀠 (t0 ) := T t 0 χ(∂ t |t 0 ) .

(13.18)

Choose some smooth chart (U, φ) on M and the corresponding coordinate basis (∂ i |χ(t 0) )si=1 , where s = dim M. Then, the coordinate representation of χ 󸀠 (t0 ) is the following: 󵄨 dχ i (t) 󵄨󵄨󵄨 󵄨󵄨 ∂ i |χ(t 0) . χ 󸀠 (t0 ) = dt 󵄨󵄨󵄨t 0 Here, χ i , i = 1, . . . , s, is the coordinate representation of χ. The tangent space to a smooth manifold with boundary. Suppose that M is a smooth s-dimensional manifold with boundary. Let p ∈ M. The space C∞ (M) of germs at p is defined in a similar manner as for the smooth manifolds without boundary. It does not matter whether p is an interior point or a boundary point. The tangent space T p M is then defined as the vector space of all derivations on C∞ (M). It has dimension s everywhere [48]. One can consider derivations of the algebra C∞ (M) similarly [3]. At the same time, the approach of the equivalence class of curves requires a minor modification. In the case of a boundary point, curves should be defined not only on open intervals but also on the intervals of the form [0, ε[ or ]−ε, 0] [3]. For every point p ∈ M, one has the coordinate basis (∂ i |p )si=1 , for T p M. Note that if p ∈ ∂M then, since the boundary ∂M is a smooth (s − 1)-dimensional manifold (submanifold of M), one can consider either the vector space T p M of dimension s or the vector space T p ∂M of dimension s − 1.

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The definitions of the tangent map and the cotangent space are repeated unchanged. We adopt the convention of identifying T p ∂M with its image under the injective map T p ι ∂M : T p ∂M → T p M, which is tangent to the inclusion map ι ∂M : ∂M → M. Inward–outward pointing vectors. Suppose that M is a smooth manifold with a boundary and p ∈ ∂M. Vectors of T p M can be divided into three classes: vectors that are tangent to T p ∂M, vectors, that are inward-pointing, and vectors that are outwardpointing. The latter two classes are defined as follows [3]. A vector u ∈ T p M \ T p ∂M is called inward-pointing, if there exists a smooth curve χ : [0, ε[→ M, ε > 0, such that χ(0) = p and χ 󸀠 (0) = u. A vector u ∈ T p M \ T p ∂M is called outward-pointing, if there exists a smooth curve χ : ]−ε, 0] → M, ε > 0, such that χ(0) = p and χ 󸀠 (0) = u.

13.2.5 The “Tower” Suppose that M is an s-dimensional smooth manifold (with or without boundary). The tangent, T p M, and the cotangent, T ∗p M, spaces are associated with each point p ∈ M. They represent the “upper” level of the “tower” and are not sufficient to construct mathematical models of continuum mechanics. Other “levels” are constructed by means of the “upper one” and the additional “building technique”, the abstract tensor product of vector spaces introduced in Chapter 12, Section 12.5.3. Such a formalism does not need a Euclidean structure and provides an effective way to construct tensors in non-Euclidean spaces. Let p ∈ M. One can generate all possible finite tensor products from copies of T p M and T ∗p M by using the notion of the abstract tensor product. Among them are (k ∈ ℕ): ∙ Contravariant tensors of rank k: T k (T p M) := ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ TpM ⊗ ⋅ ⋅ ⋅ ⊗ Tp M , k copies

which can be decomposed over the basis (e i1 ⊗ ⋅ ⋅ ⋅ ⊗ e i k )1≤i 1 ,...,i k ≤s . ∙

Here, (e i )si=1 is a basis for T p M. Covariant tensors of rank k: T ∗p M ⊗ ⋅ ⋅ ⋅ ⊗ T ∗p M , T k (T ∗p M) := ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ k copies

which can be decomposed over the basis (ϑ i1 ⊗ ⋅ ⋅ ⋅ ⊗ ϑ i k )1≤i 1 ,...,i k ≤s . Here, (ϑ i )si=1 is a basis for T ∗p M.

13.2 The “Tower” of Tensor Spaces |

∙

337

Mixed tensors of type (k, l): T p M ⊗ ⋅ ⋅ ⋅ ⊗ T p M ⊗ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ T ∗p M ⊗ ⋅ ⋅ ⋅ ⊗ T ∗p M . T (k,l)(T p M) := ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ k copies

l copies

The basis in this space of mixed tensors can be combined as follows: (e i1 ⊗ ⋅ ⋅ ⋅ ⊗ e i k ⊗ ϑ j1 ⊗ ⋅ ⋅ ⋅ ⊗ ϑ j l )1≤i

1 ,...,i k ≤s

,

,

1≤j1 ,...,j l ≤s

where (e i )si=1 is a basis for T p M and (ϑ i )si=1 is the dual basis for T ∗p M. We use the standard agreements: T 0 (T p M) = T 0 (T ∗p M) := ℝ , T (0,k) (T p M) = T k (T ∗p M) ,

T (k,0) (T p M) = T k (T p M) ,

...

The dimensions of the spaces introduced above are: dim T k (T p M) = dim T k (T ∗p M) = s k and dim T (k,l)(T p M) = s k+l . All such spaces can be ordered by the levels of the symbolic “tower”, as is shown below: ...

...

Tp M ⊗ Tp M

T p M ⊗ T ∗p M Tp M

≅

...

...

T ∗p M ⊗ T p M

T ∗p M ⊗ T ∗p M

T ∗p M

13.2.6 Exterior Forms The calculus of exterior forms was developed in the papers of F. Klein [41], H. Weyl [35], and E. Cartan [150] for a mathematically rigorous definition of the operations of integration on manifolds that are of general structure. Since the mid-twentieth century, this calculus has been intensively used in theoretical physics to formalize the fields defined in the curved space-time [34, 151]. In continuum mechanics, the calculus of exterior forms is used rather rarely: most of the problems are stated in Euclidean space, and there is no need to complicate the traditional apparatus of integration. At the same time, the present book goes beyond traditional Euclidean statements, and it seems most concise and convenient to use the elements of the calculus of exterior forms to determine the fields that represent the densities of physical quantities: mass forces, stresses, etc. Exterior forms: definition. Let M be a smooth s-dimensional manifold. By virtue of the natural isomorphism it is convenient to consider each element T ∈ T k (T ∗p M) as a

338 | 13 Review of Smooth Manifolds and Vector Bundles

k-linear mapping T : ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ TpM × ⋅ ⋅ ⋅ × Tp M → ℝ . k times

T k (T ∗p M)

A mapping ω ∈ is called an exterior k-form (or antisymmetric tensor) [3] if for all tangent vectors u 1 , . . . , u k ∈ T p M and for each pair of distinct indices i, j ∈ {1, . . . , k}, one has ω(u 1 , . . . , u i , . . . , u j , . . . , u k ) = −ω(u 1 , . . . , u j , . . . , u i , . . . , u k ) . The set Λ k (T ∗p M) ⊂ T k (T ∗p M) of all exterior forms is the subspace of T k (T ∗p M). The wedge product. The tensor product ⊗, applied to exterior forms, in general does not lead beyond the set of exterior forms, but a new operation, namely, the wedge (exterior) product can be defined in such a way. Recall that a permutation is a bijection σ : {1, . . . , k} → {1, . . . , k}, k ∈ ℕ. Let S k be the set of all permutations. This set, being endowed with a binary operation of the composition ∘, is a group. For any tensor T ∈ T k (T ∗p M), a permutation σ ∈ S k generates a new tensor σ T ∈ T k (T ∗p M): ∀u 1 , . . . , u k ∈ T p M : σ T(u 1 , . . . , u k ) := T(u σ(1) , . . . , u σ(k) ) . For a permutation σ ∈ S k , its sign is the number {1, sgn σ := { −1, {

if σ is even , if σ is odd .

One has the equivalent definition for the exterior form as follows: ω ∈ Λ k (T ∗p M) iff ω ∈ T k (T ∗p M), and for all tangent vectors u 1 , . . . , u k ∈ T p M, for every permutation σ ∈ S k , one has ω(u σ(1) , . . . , u σ(k) ) = (sgn σ)ω(u 1 , . . . , u k ) . The mapping Alt : T k (T ∗p M) → Λ k (T ∗p M), Alt T :=

1 ∑ (sgn σ)(σ T) , k! σ∈S k

is called an alternation [3]. In the explicit form, for all u 1 , . . . , u k ∈ T p M, one has Alt T(u 1 , . . . , u k ) =

1 ∑ (sgn σ)T(u 1 , . . . , u k ) . k! σ∈S k

Λ k (T ∗p M)

Λ l (T ∗p M).

and η ∈ There are several definitions for the wedge Let ω ∈ product. The first one is ω ∧ η := Alt (ω ⊗ η). Such a definition is used in [145]. We prefer to use another definition: ω ∧ η :=

(k + l)! Alt (ω ⊗ η) . k!l!

(13.19)

13.2 The “Tower” of Tensor Spaces | 339

Remark 13.1. In particular, for covectors α and β, one has α∧β=α⊗β−β⊗α. Hence, ∧ : Λ k (T ∗p M) × Λ l (T ∗p M) → Λ k+l (T ∗p M). We refer to such a mapping as the wedge product or exterior product. It satisfies the following properties [3]: ∧1 ) Bilinearity. For ω, ω󸀠 ∈ Λ k (T ∗p M), η, η󸀠 ∈ Λ l (T ∗p M) and a, a󸀠 ∈ ℝ, (aω + a󸀠 ω󸀠 ) ∧ η = a(ω ∧ η) + a󸀠 (ω󸀠 ∧ η) , ω ∧ (aη + a󸀠 η󸀠 ) = a(ω ∧ η) + a󸀠 (ω ∧ η 󸀠 ) . ∧2 ) Associativity. For ω ∈ Λ k (T ∗p M), η ∈ Λ l (T ∗p M) and μ ∈ Λ r (T ∗p M), ω ∧ (η ∧ μ) = (ω ∧ η) ∧ μ . ∧3 ) Anticommutativity. For ω ∈ Λ k (T ∗p M), η ∈ Λ l (T ∗p M), ω ∧ η = (−1)kl η ∧ ω . ∧4 ) If ν1 , . . . , ν k ∈ T ∗p M and u 1 , . . . , u k ∈ T p M, then ν1 ∧ ⋅ ⋅ ⋅ ∧ ν k (u 1 , . . . , u k ) = det[ν i (u j )] . The factor (k + l)!/(k!l!) is essential for ∧4 ). In this regard, in [3] the term “determinant convention” is used for (13.19). Suppose that (ϑ i )si=1 is a basis for T ∗p M. The collection (ϑ i1 ∧ ⋅ ⋅ ⋅ ∧ ϑ i k )1≤i 1